Free math worksheets

- Each expression contains like terms. - Combine the numerical coefficients. - Keep track of each coefficient’s sign.

1. \(12-7+3=8\), so the result is \(8a\). 2. \(-5+9-6=-2\), so the result is \(-2x^2\). 3. \(4.5-1.2-2.1=1.2\), so the result is \(1.2y\).

1) \(8a\) 2) \(-2x^2\) 3) \(1.2y\)

- Subtracting a negative term changes to addition. - Combine the signed coefficients. - Opposite terms cancel.

1. \(12x^2-(-8x^2)=12x^2+8x^2=20x^2\). 2. \(-15y^3-(-10y^3)=-15y^3+10y^3=-5y^3\). 3. \(7a^4-(+11a^4)=7a^4-11a^4=-4a^4\). 4. \(-b^2-(-b^2)=-b^2+b^2=0\).

1) \(20x^2\) 2) \(-5y^3\) 3) \(-4a^4\) 4) \(0\)

- Rewrite a variable in a denominator using exponent notation. - Polynomial exponents on the variable must be nonnegative integers. - The degree comes from the term with the greatest allowed exponent.

1. \(P(x)\) is a polynomial because its exponents on \(x\) are nonnegative integers. 2. Its highest exponent is \(3\), so its degree is \(3\), and the coefficient of the leading term \(4x^3\) is \(4\). 3. \(Q(x)\) is not a polynomial in \(x\) because \(\frac{5}{x}=5x^{-1}\) has a negative exponent.

\(P(x)\) is the polynomial. Degree: \(3\) Leading coefficient: \(4\)

- Compare the variable part of each term, not just its coefficient. - Like terms must have matching exponents.

1. Like terms must have the same variable part with the same exponent. 2. The terms \(4x^2\) and \(5x^2\) both have variable part \(x^2\), so they are like terms.

\(4x^2\) and \(5x^2\)

- Remove parentheses before combining terms. - Combine only like terms. - Check the signs carefully, especially when subtracting parentheses.

1. For a), remove the parentheses and combine like terms: \(9a - 4a + 7b - 2b = 5a + 5b\). 2. For b), distribute and combine like terms: \(4x(3 - y) + 4xy - 10x = 12x - 4xy + 4xy - 10x = 2x\).

a) \(5a + 5b\) b) \(2x\)

- When subtracting parentheses, change the sign of every term inside. - Identify like terms by matching both their variables and exponents. - Expand all products before combining terms. - A negative factor times a negative term gives a positive term.

1. For a), \(15x - 4y + 10 - 6x - 2y + 3 = 9x - 6y + 13\). 2. For b), \(4.2a^2 - 2a + 1.8a^2 + 5a = 6a^2 + 3a\). 3. For c), \(12 - 3x + 12 + 10 - 4x = 34 - 7x\).

a) \(9x - 6y + 13\) b) \(6a^2 + 3a\) c) \(34 - 7x\)

- Simplify each product before subtracting. - Recognize the conjugate product \((a - 4)(a + 4)\). - Distribute the negative sign across the second result.

1. Distribute: \(3a(2a - 5) = 6a^2 - 15a\). 2. Use the difference of squares: \((a - 4)(a + 4) = a^2 - 16\). 3. Subtract and combine like terms: \(6a^2 - 15a - (a^2 - 16) = 5a^2 - 15a + 16\).

\(5a^2 - 15a + 16\)

- Expand both binomial squares. - Apply the subtraction sign to every term in the second polynomial. - Combine like terms.

1. Expand the first square: \((x - 5)^2 = x^2 - 10x + 25\). 2. Expand the second square: \((x + 2)^2 = x^2 + 4x + 4\). 3. Subtract: \(x^2 - 10x + 25 - x^2 - 4x - 4 = -14x + 21\).

\(-14x + 21\)

- Which exponent property applies when a power is raised to another power? - What happens to the exponents when powers with the same base are divided? - Pay close attention to the signs when distributing \(-3x\).

1. Simplify the power in the numerator: \((2x^3)^2 = 2^2(x^3)^2 = 4x^6\). 2. Divide powers with the same base: \(\frac{4x^6}{x^4} = 4x^{6-4} = 4x^2\). 3. Distribute: \(-3x(x - 2) = -3x^2 + 6x\). 4. Combine like terms: \(4x^2 - 3x^2 + 6x = x^2 + 6x\).

\(x^2 + 6x\)

- Match each expression to a binomial identity. - Square every factor in a monomial. - Identify the two quantities that play the roles of \(A\) and \(B\).

1. a) \((3a)^2 + 2(3a)(5b) + (5b)^2 = 9a^2 + 30ab + 25b^2\). 2. b) \((4x^2)^2 - 2(4x^2)(7) + 7^2 = 16x^4 - 56x^2 + 49\). 3. c) Use the difference of squares: \((2y^3)^2 - (3x^2)^2 = 4y^6 - 9x^4\).

a) \((3a)^2+2(3a)(5b)+(5b)^2=9a^2+30ab+25b^2\) b) \((4x^2)^2-2(4x^2)(7)+7^2=16x^4-56x^2+49\) c) \((2y^3)^2-(3x^2)^2=4y^6-9x^4\)

- Expand each binomial expression separately. - Apply the subtraction sign to every term in part a). - Combine like terms after expansion.

1. a) Expand both squares: \(4x^2 + 20x + 25 - (4x^2 - 20x + 25) = 40x\). 2. b) Expand the square and use the difference of squares: \(9y^2 - 12y + 4 + 9y^2 - 4 = 18y^2 - 12y\).

a) \(40x\) b) \(18y^2 - 12y\)

- Identify the binomial identity for each product or square. - Distribute the subtraction sign in part a). - Combine like terms after expansion.

1. a) Expand both squares: \(x^2 + 8x + 16 - (x^2 - 8x + 16) = 16x\). 2. b) Use the difference of squares and expand the binomial square: \(4a^2 - 9 + a^2 + 6a + 9 = 5a^2 + 6a\).

a) \(16x\) b) \(5a^2 + 6a\)

- When multiplying powers with the same base, add the exponents. - Pay close attention to the sign when the outside factor is negative. - Multiply the outside monomial by every term in the polynomial.

1. For a), distribute and add exponents of like bases: \(2x^3(4x^2 - 5x + 1) = 8x^5 - 10x^4 + 2x^3\). 2. For b), \(-ab^2(3a^2b - 2b + 7) = -3a^3b^3 + 2ab^3 - 7ab^2\). 3. For c), \(4s^2t^3(s^2 - 3st + 2t^2) = 4s^4t^3 - 12s^3t^4 + 8s^2t^5\).

a) \(8x^5 - 10x^4 + 2x^3\) b) \(-3a^3b^3 + 2ab^3 - 7ab^2\) c) \(4s^4t^3 - 12s^3t^4 + 8s^2t^5\)

- Compare both the variables and their exponents when identifying like terms. - Rewrite the fractional coefficients with a common denominator. - Combine each group of like terms separately.

1. The \(x^2y\)-terms and \(xy^2\)-terms are not like terms, so combine them separately. 2. Combine the \(x^2y\)-terms: \(2.4x^2y+0.6x^2y=3x^2y\). 3. Combine the \(xy^2\)-terms: \(-\frac{3}{5}+\frac{1}{10}=-\frac{6}{10}+\frac{1}{10}=-\frac{5}{10}=-0.5\). 4. The simplified expression is \(3x^2y-0.5xy^2\).

\(3x^2y-0.5xy^2\)

- An \(a^3\)-term with no visible coefficient has coefficient \(1\). - Convert fractions and decimals to a common form. - Combine terms only when their variable parts and exponents match.

1. Combine the \(a^3\)-terms: \(\frac{2}{3}+\frac{1}{6}-1=\frac{4}{6}+\frac{1}{6}-\frac{6}{6}=-\frac{1}{6}\). 2. Combine the \(a^2\)-terms: \(-0.75+\frac{1}{4}=-0.75+0.25=-0.5\). 3. The simplified polynomial is \(-\frac{1}{6}a^3-0.5a^2\).

\(-\frac{1}{6}a^3-0.5a^2\)

- Like terms have the same variables raised to the same powers. - Combine the coefficients while keeping their signs. - The constant term remains separate.

1. Combine the \(x^2\)-terms: \(0.7x^2+1.3x^2=2x^2\). 2. Combine the \(x\)-terms: \(-5x+2x=-3x\). 3. Keep the constant term \(-11\). The simplified polynomial is \(2x^2-3x-11\).

\(2x^2-3x-11\)

- Each part contains like terms with the same variable and exponent. - Convert mixed numbers to decimals or improper fractions. - A term disappears when its coefficient is zero.

1. For part a, convert \(1\frac{4}{5}\) to \(1.8\). Then \(4.2-6+1.8=0\), so the expression equals \(0\). 2. For part b, convert \(2\frac{3}{4}\) to \(2.75\). Then \(2.75-1.5-3=-1.75\), so the result is \(-1.75x^2\).

a) \(0\) b) \(-1.75x^2\), or \(-1\frac{3}{4}x^2\)

- Use a common denominator to combine fractional coefficients. - Keep track of every negative sign. - Simplify the final coefficient when possible.

1. For part a, use denominator \(12\): \(\frac{7}{12}-\frac{3}{12}-\frac{8}{12}=-\frac{4}{12}=-\frac{1}{3}\). The result is \(-\frac{1}{3}a^3\). 2. For part b, use denominator \(8\): \(-\frac{4}{8}+\frac{5}{8}-\frac{6}{8}=-\frac{5}{8}\). The result is \(-\frac{5}{8}b^4\).

a) \(-\frac{1}{3}a^3\) b) \(-\frac{5}{8}b^4\)

- Each group consists of like terms. - Combine only the coefficients. - Remember that coefficients of \(1\), \(-1\), and \(0\) have special written forms.

1. \(4.2-5.8+0.6=-1\), so the result is \(-x^2\). 2. \(-0.75+1.5-0.25=0.5\), so the result is \(0.5ab^2\). 3. \(12-15-3+7=1\), so the result is \(y^5\). 4. \(0.15-0.4+0.25=0\), so the result is \(0\).

1) \(-x^2\) 2) \(0.5ab^2\) 3) \(y^5\) 4) \(0\)

- Combine the known like terms first. - Treat the missing value as an unknown in an equation. - Use inverse operations on the coefficients.

1. Let the missing coefficient be \(k\). The known coefficients add to \(2.5+1.2=3.7\), so \(3.7-k=0.7\). Thus \(k=3\), and the missing term is \(3a^3b\). 2. Let the missing coefficient be \(c\). Combine the known coefficients: \(3.4-5=-1.6\). Then \(c-1.6=-2.1\), so \(c=-0.5\).

1) \(3a^3b\) 2) \(-0.5\)

- Like terms must have identical variable parts and exponents. - Use the commutative property to recognize that \(ab=ba\). - Combine constant terms separately.

1. Combine the \(m\)-terms and \(n\)-terms: \(10m-4m=6m\) and \(-3n+7n=4n\). The result is \(6m+4n\). 2. Combine terms with matching powers: \(2x^2-7x^2=-5x^2\) and \(5x-x=4x\). The result is \(-5x^2+4x\). 3. Since \(ba=ab\), \(12ab-5ba=7ab\). The term \(4a\) is not like an \(ab\)-term, so the result is \(7ab+4a\). 4. Combine the \(z\)-terms and constants: \(\frac{3}{8}z-\frac{1}{8}z=\frac{1}{4}z\) and \(\frac{1}{4}+\frac{3}{4}=1\). The result is \(\frac{1}{4}z+1\).

1) \(6m+4n\) 2) \(-5x^2+4x\) 3) \(7ab+4a\) 4) \(\frac{1}{4}z+1\)

- Group terms with the same variable parts and exponents. - Keep each term’s sign with its coefficient. - Terms with different powers cannot be combined.

1. \(18x^2-5x^2=13x^2\), and \(7x-7x=0\). The result is \(13x^2\). 2. \(-4a-3a=-7a\), and \(-9+2=-7\). The result is \(-7a-7\). 3. \(5y^3-3y^3=2y^3\), and \(-2y+4y=2y\). The result is \(2y^3+2y\). 4. \(-2m^2+5m^2=3m^2\), and \(-n-4n=-5n\). The result is \(3m^2-5n\).

1) \(13x^2\) 2) \(-7a-7\) 3) \(2y^3+2y\) 4) \(3m^2-5n\)

- Combine coefficients only when the variable parts match exactly. - Use the commutative property to compare \(pq\) and \(qp\). - Terms can cancel when their coefficients are opposites.

1. Combine by power: \(3.5z^2+1.5z^2=5z^2\) and \(-1.2z-0.8z=-2z\). The result is \(5z^2-2z\). 2. \(\frac{2}{5}b+\frac{3}{5}b=b\), and \(-\frac{1}{2}c-\frac{3}{2}c=-2c\). The result is \(b-2c\). 3. Since \(qp=pq\), \(9pq+2qp=11pq\), and \(-4p-p=-5p\). The result is \(11pq-5p\). 4. \(x^4+2x^4=3x^4\), and the \(x^2\)-terms cancel. The result is \(3x^4\).

1) \(5z^2-2z\) 2) \(b-2c\) 3) \(11pq-5p\) 4) \(3x^4\)

- Like terms have exactly the same variables and exponents. - A variable term with no visible coefficient has coefficient \(1\). - Combine each group of coefficients separately.

1. Combine the \(a\)-terms and \(b\)-terms: \(9a+3a+a=13a\) and \(-4b-7b=-11b\). The result is \(13a-11b\). 2. Combine by power: \(1.2x^2+0.8x^2-x^2=x^2\), and \(-0.5x+1.5x=x\). The result is \(x^2+x\). 3. Combine each variable part separately: \(\frac{3}{8}u^2v+\frac{5}{8}u^2v=u^2v\), and \(-\frac{1}{2}uv^2+\frac{3}{4}uv^2=\frac{1}{4}uv^2\). The result is \(u^2v+\frac{1}{4}uv^2\).

1) \(13a-11b\) 2) \(x^2+x\) 3) \(u^2v+\frac{1}{4}uv^2\)

- Compare the complete variable parts, including exponents. - A group of like terms cancels when its coefficients add to zero. - Keep constant terms separate.

1. Combine the \(x^2y\)-terms and \(xy\)-terms: \(17+3=20\) and \(-8-2+1=-9\). The result is \(20x^2y-9xy\). 2. The \(a^3b\)-coefficients add to \(2.5+0.5-3=0\), and the \(ab^3\)-coefficients add to \(-1.4+1.4=0\). The expression simplifies to \(0\). 3. The \(z^2\)-coefficients add to \(\frac{2}{3}+\frac{1}{3}=1\), and the \(z\)-coefficients add to \(-\frac{1}{4}-\frac{3}{4}=-1\). The result is \(z^2-z+5\).

1) \(20x^2y-9xy\) 2) \(0\) 3) \(z^2-z+5\)

- Group terms with identical variables and exponents. - Use a common denominator for fractional coefficients. - Keep each coefficient’s sign.

1. Group the \(x^2y\)-terms and \(xy^2\)-terms separately. 2. For \(x^2y\), \(\frac{1}{3}+\frac{5}{6}-\frac{1}{6}=\frac{2}{6}+\frac{5}{6}-\frac{1}{6}=1\). 3. For \(xy^2\), \(-\frac{5}{2}-\frac{1}{4}=-\frac{10}{4}-\frac{1}{4}=-\frac{11}{4}\). 4. The simplified expression is \(x^2y-\frac{11}{4}xy^2\).

\(x^2y-\frac{11}{4}xy^2\)

- \(a^2\) and \(ab\) are different variable parts. - The coefficient of \(-a^2\) is \(-1\). - Combine each group of coefficients separately.

1. Combine the \(a^2\)-terms: \(2.4+0.6-1=2\). 2. Combine the \(ab\)-terms: \(-1.5+3=1.5\). 3. The simplified expression is \(2a^2+1.5ab\).

\(2a^2+1.5ab\)

- Combine the signed coefficients of the like terms. - Rewrite fractions with a common denominator. - A term with coefficient zero disappears.

1. \(14-8-6=0\), so the expression equals \(0\). 2. \(-4.2-1.5+0.7=-5\), so the result is \(-5xy\). 3. \(\frac{3}{4}-\frac{1}{2}+\frac{1}{4}=\frac{3}{4}-\frac{2}{4}+\frac{1}{4}=\frac{1}{2}\), so the result is \(\frac{1}{2}b^2\).

1) \(0\) 2) \(-5xy\) 3) \(\frac{1}{2}b^2\)

- Group terms by their powers of \(x\). - Keep each term’s sign. - Like terms must have identical variable parts and exponents.

1. Combine the quadratic terms: \(5x^2-8x^2=-3x^2\). 2. Combine the linear terms: \(-12x+4x=-8x\). 3. Combine the constants: \(9-15=-6\). 4. In descending powers, the result is \(-3x^2-8x-6\). 5. The variable parts \(x^2\) and \(x\) have different exponents, so they are not like terms and cannot be combined by adding their coefficients.

a) \(-3x^2-8x-6\) b) \(x^2\)-terms and \(x\)-terms have different exponents, so they are not like terms.

- Add fractions by using a common denominator. - A variable term with no visible coefficient has coefficient \(1\). - Adding like terms does not change their exponents.

1. Statement 1 is correct because \(5-8=-3\). 2. Statement 2 is incorrect. Use a common denominator: \(\frac{1}{2}x+\frac{1}{4}x=\frac{2}{4}x+\frac{1}{4}x=\frac{3}{4}x\). The original work incorrectly added the denominators. 3. Statement 3 is correct because \(-0.3+1=0.7\). 4. Statement 4 is incorrect. Adding like terms changes the coefficient, not the exponent: \(b^2+b^2=2b^2\). 5. Statement 5 is correct because \(4-1=3\).

1) Correct. 2) Incorrect; \(\frac{3}{4}x\). 3) Correct. 4) Incorrect; \(2b^2\). 5) Correct.

- Combine the coefficients separately from the variable parts. - An unwritten coefficient is \(1\). - Use a common denominator for fractional coefficients.

1. For part a, \(-1.2+2.5-0.8=0.5\), so the result is \(0.5k\). 2. For part b, \(\frac{2}{3}-\frac{1}{6}-1=\frac{4}{6}-\frac{1}{6}-\frac{6}{6}=-\frac{1}{2}\), so the result is \(-\frac{1}{2}z\). 3. For part c, \(1.5-2+0.5=0\), so the expression equals \(0\). 4. For part d, \(\frac{3}{4}+\frac{1}{8}-1=\frac{6}{8}+\frac{1}{8}-\frac{8}{8}=-\frac{1}{8}\), so the result is \(-\frac{1}{8}ab\).

a) \(0.5k\) b) \(-\frac{1}{2}z\) c) \(0\) d) \(-\frac{1}{8}ab\)

- Confirm that every term has the same variables and exponents. - Rewrite additions of negative terms as subtraction. - Combine the signed coefficients.

1. All terms have the same variable part \(ab^2\). 2. Combine the coefficients: \(12-5-8-2=-3\). 3. The simplified expression is \(-3ab^2\).

\(-3ab^2\)

- Group terms with identical powers of \(x\). - Keep each coefficient’s sign. - Combine the cubic and linear terms separately.

1. Combine the \(x^3\)-terms: \(0.7-0.5-0.4=-0.2\). 2. Combine the \(x\)-terms: \(-1.2+2.5=1.3\). 3. The simplified expression is \(-0.2x^3+1.3x\).

\(-0.2x^3+1.3x\)

- Group terms with matching variables and exponents. - Combine constants separately. - Keep the signs attached to their terms.

1. Combine the \(x^2\)-terms: \(4.8x^2-1.3x^2=3.5x^2\). 2. Combine the \(x\)-terms: \(-2.5x-0.5x=-3x\). 3. Combine the constants: \(3.7-1.2=2.5\). 4. The simplified expression is \(3.5x^2-3x+2.5\).

\(3.5x^2-3x+2.5\)

- \(u^2v\) and \(uv^2\) are not like terms. - Use common denominators for the fractional coefficients. - Keep each negative sign with its coefficient.

1. Combine the \(u^2v\)-terms: \(-\frac{4}{9}-\frac{1}{3}=-\frac{4}{9}-\frac{3}{9}=-\frac{7}{9}\). 2. Combine the \(uv^2\)-terms: \(\frac{1}{2}-\frac{3}{10}+\frac{1}{5}=\frac{5}{10}-\frac{3}{10}+\frac{2}{10}=\frac{2}{5}\). 3. The simplified expression is \(-\frac{7}{9}u^2v+\frac{2}{5}uv^2\).

\(-\frac{7}{9}u^2v+\frac{2}{5}uv^2\)

- Group terms with identical variable parts. - Combine the signed decimal coefficients. - Keep the three variable parts separate.

1. Combine the \(m^2\)-terms: \(7.5-5.8=1.7\). 2. Combine the \(mn\)-terms: \(-4.2+2.3=-1.9\). 3. Combine the \(n^2\)-terms: \(1.6-3.1=-1.5\). 4. The simplified expression is \(1.7m^2-1.9mn-1.5n^2\).

\(1.7m^2-1.9mn-1.5n^2\)

- A plus sign before parentheses leaves the signs inside unchanged. - Combine only terms with matching variables and exponents. - Combine constants separately.

1. Combine matching variable terms: \(12a+5a=17a\) and \(7b-9b=-2b\). The result is \(17a-2b\). 2. Combine by degree: \(4x^2+2x^2=6x^2\), \(-3x+5x=2x\), and \(2-6=-4\). The result is \(6x^2+2x-4\). 3. Combine the \(y\)-terms and constants: \(2.4y+0.6y=3y\) and \(-1.5+3.8=2.3\). The result is \(3y+2.3\).

1) \(17a-2b\) 2) \(6x^2+2x-4\) 3) \(3y+2.3\)

- Combine only terms with identical variables and exponents. - Opposite terms cancel. - Group each variable part before adding coefficients.

1. Combine matching variable parts: \(5r^2s-4r^2s=r^2s\) and \(-2rs^2+3rs^2=rs^2\). The result is \(r^2s+rs^2\). 2. The \(a^3\)-terms cancel, \(4a^2+2a^2=6a^2\), and the constants cancel. The result is \(6a^2\). 3. \(2x+x=3x\), \(3y-5y+2y=0\), and \(-4z+2z+2z=0\). The result is \(3x\).

1) \(r^2s+rs^2\) 2) \(6a^2\) 3) \(3x\)

- Group terms with matching variable parts. - Use common denominators for fractional coefficients and constants. - Opposite terms cancel.

1. \((9y-12)+(4y+15)=13y+3\). 2. \((-3a+8b)+(3a-10b)=-2b\). 3. \((2.5x^2-4x)+(-1.5x^2+4x-7)=x^2-7\). 4. \(\left(\frac{1}{2}k+\frac{2}{3}\right)+\left(\frac{1}{4}k-\frac{1}{6}\right)=\frac{3}{4}k+\frac{1}{2}\).

1) \(13y+3\) 2) \(-2b\) 3) \(x^2-7\) 4) \(\frac{3}{4}k+\frac{1}{2}\)

- Group terms with identical variable parts. - Add the signed decimal coefficients. - Keep \(a^2\), \(ab\), and \(b^2\) separate.

1. Combine the \(a^2\)-coefficients: \(7.2-2.8+0.6=5\). 2. Combine the \(ab\)-coefficients: \(-3.5+4.2-1.1=-0.4\). 3. Combine the \(b^2\)-coefficients: \(1.4-0.9-1.2=-0.7\). 4. The sum is \(5a^2-0.4ab-0.7b^2\).

\(5a^2-0.4ab-0.7b^2\)

- Treat each blank as an unknown expression. - Pay attention to subtraction of negative terms. - Substitute each result to verify it.

1. Let the missing expression be \(M\). Then \(M=4.5z^2-7z^2=-2.5z^2\). Check: \(4.5z^2-(-2.5z^2)=7z^2\). 2. Let the missing expression be \(N\). Then \(N+3x^3=-x^3\), so \(N=-4x^3\). 3. Combine like terms: \(-1.2a-0.8a=-2a\).

1) \(-2.5z^2\) 2) \(-4x^3\) 3) \(-2a\)

- Combine the numerical coefficients. - Subtracting a negative coefficient changes to addition. - Keep the variable part unchanged.

1. \(5.4-2.1=3.3\), so the result is \(3.3k^2\). 2. \(-3.8-(-5.2)=-3.8+5.2=1.4\), so the result is \(1.4x^4\). 3. \(0.25-0.75=-0.5\), so the result is \(-0.5m^2n\).

1) \(3.3k^2\) 2) \(1.4x^4\) 3) \(-0.5m^2n\)

- Convert fractional and decimal coefficients to a common form. - \(a^2b\) and \(ab^2\) are different variable parts. - A coefficient of \(-1\) is written as a minus sign.

1. Convert \(\frac{3}{4}\) to \(0.75\): \(0.75-0.5-1.25=-1\). The result is \(-y^3\). 2. Combine the \(a^2b\)-terms: \(6.7-4.7=2\). Combine the \(ab^2\)-terms: \(-2.3+0.3=-2\). The result is \(2a^2b-2ab^2\).

1) \(-y^3\) 2) \(2a^2b-2ab^2\)

- Combine only terms with identical variable parts. - Subtracting a negative term changes to addition. - Use common denominators for fractional coefficients.

1. \(12-15+4=1\), so the result is \(xy\). 2. \(-5-(-8)-3=-5+8-3=0\), so the expression equals \(0\). 3. \(4.5+1.5-7=-1\), so the result is \(-z\). 4. \(\frac{1}{2}+\frac{3}{2}=2\), so the result is \(2x\).

1) \(xy\) 2) \(0\) 3) \(-z\) 4) \(2x\)

- A negative sign before parentheses reverses every sign inside. - Combine only matching variable parts. - Keep \(u^2\)-terms separate from \(u\)-terms.

1. For part a, distribute the negative sign: \(9k-4k-3m=5k-3m\). 2. For part b, distribute the negative sign and combine like terms: \(2x-8y-5x+3y=-3x-5y\). 3. For part c, \(15u^2-7u^2+10u-4u=8u^2+6u\).

a) \(5k-3m\) b) \(-3x-5y\) c) \(8u^2+6u\)

- Reverse every sign in the expression being subtracted. - Group terms with identical variable parts. - Check whether any terms cancel.

1. For part a, distribute the negative sign: \(6x-9y+4z-15x-3y+8z\). 2. Combine like terms: \((6-15)x+(-9-3)y+(4+8)z=-9x-12y+12z\). 3. For part b, distribute the negative sign: \(4a^2-ab+2b^2-4a^2-5ab+3b^2\). 4. Combine like terms: the \(a^2\)-terms cancel, \(-ab-5ab=-6ab\), and \(2b^2+3b^2=5b^2\).

a) \(-9x-12y+12z\) b) \(-6ab+5b^2\)

- Reverse every sign in the second polynomial. - Combine terms with identical variable parts. - Use common denominators for fractional coefficients.

1. Distribute the negative sign: \(\frac{3}{10}x^2-\frac{2}{3}xy+\frac{5}{6}y^2-\frac{1}{5}x^2-\frac{1}{4}xy+\frac{1}{2}y^2\). 2. Combine the \(x^2\)-terms: \(\frac{3}{10}-\frac{1}{5}=\frac{1}{10}\). 3. Combine the \(xy\)-terms: \(-\frac{2}{3}-\frac{1}{4}=-\frac{8}{12}-\frac{3}{12}=-\frac{11}{12}\). 4. Combine the \(y^2\)-terms: \(\frac{5}{6}+\frac{1}{2}=\frac{4}{3}\). 5. The result is \(\frac{1}{10}x^2-\frac{11}{12}xy+\frac{4}{3}y^2\).

\(\frac{1}{10}x^2-\frac{11}{12}xy+\frac{4}{3}y^2\)

- Reverse every sign in the second expression. - Group terms with matching variable products. - Combine the signed decimal coefficients.

1. Distribute the negative sign: \(4.7xy-2.3yz+5xz-1.2xy-3.7yz+2xz\). 2. Combine the \(xy\)-terms: \(4.7xy-1.2xy=3.5xy\). 3. Combine the \(yz\)-terms: \(-2.3yz-3.7yz=-6yz\). 4. Combine the \(xz\)-terms: \(5xz+2xz=7xz\). 5. The result is \(3.5xy-6yz+7xz\).

\(3.5xy-6yz+7xz\)

- Convert all coefficients to decimals first. - Reverse every sign in the second polynomial. - Combine the \(x^2\), \(xy\), and \(y^2\) terms separately.

1. Convert the fractional coefficients: \(4\frac{1}{2}=4.5\), \(\frac{4}{5}=0.8\), and \(3\frac{1}{2}=3.5\). 2. Distribute the negative sign: \(4.5x^2-0.3xy+2y^2-1.75x^2-0.8xy+3.5y^2\). 3. Combine like terms: \((4.5-1.75)x^2+(-0.3-0.8)xy+(2+3.5)y^2\). 4. The result is \(2.75x^2-1.1xy+5.5y^2\).

\(2.75x^2-1.1xy+5.5y^2\)

- Convert the fraction and mixed number to decimals. - Reverse every sign in the second polynomial. - Check whether the constant terms cancel.

1. Convert the fractional values: \(\frac{3}{8}=0.375\) and \(1\frac{1}{5}=1.2\). 2. Distribute the negative sign: \(0.6z^2-0.375z+1.2+1.4z^2-0.125z-1.2\). 3. Combine like terms: \((0.6+1.4)z^2+(-0.375-0.125)z+(1.2-1.2)\). 4. The result is \(2z^2-0.5z\).

\(2z^2-0.5z\)

- Convert mixed numbers to improper fractions. - Reverse every sign in the expression being subtracted. - Use common denominators before combining coefficients.

1. Convert \(1\frac{3}{4}\) to \(\frac{7}{4}\), distribute the negative sign, and combine like terms: \(\left(\frac{7}{4}-\frac{1}{2}\right)x+\left(-\frac{2}{5}-\frac{1}{10}\right)y=\frac{5}{4}x-\frac{1}{2}y\). 2. Distribute the negative sign and combine like terms: \(\left(\frac{2}{3}-\frac{1}{6}\right)a^2+\left(-\frac{1}{4}-\frac{3}{4}\right)ab+\left(\frac{1}{2}+1\right)b^2\). 3. The second result is \(\frac{1}{2}a^2-ab+\frac{3}{2}b^2\).

1) \(\frac{5}{4}x-\frac{1}{2}y\) 2) \(\frac{1}{2}a^2-ab+\frac{3}{2}b^2\)

- Begin with the innermost parentheses. - Combine like terms inside the brackets before removing them. - Reverse every sign when subtracting the bracketed expression.

1. Remove the inner parentheses: \(10a^2-[4a^2+6a-a^2+2a]\). 2. Combine like terms inside the brackets: \(10a^2-[3a^2+8a]\). 3. Distribute the negative sign outside the brackets: \(10a^2-3a^2-8a\). 4. Combine the quadratic terms to obtain \(7a^2-8a\).

\(7a^2-8a\)

- Reverse every sign in the polynomial being subtracted. - A plus sign before parentheses leaves the signs unchanged. - Group terms by degree.

1. Remove the parentheses: \(4x^2-3x+5-2x^2-5x+7+x^2-4x-1\). 2. Combine the quadratic terms: \((4-2+1)x^2=3x^2\). 3. Combine the linear terms: \((-3-5-4)x=-12x\). 4. Combine the constants: \(5+7-1=11\). 5. The result is \(3x^2-12x+11\).

\(3x^2-12x+11\)

- Reverse every sign in the polynomial being subtracted. - Group terms whose variable parts, including exponents, are identical. - Combine only the coefficients of like terms.

1. Distribute the negative sign before the second polynomial: \(4.5x^2y-2.3xy^2+1.2x^2y^2-1.8x^2y-3.7xy^2+0.5x^2y^2\). 2. Combine the \(x^2y\) terms: \((4.5-1.8)x^2y=2.7x^2y\). 3. Combine the \(xy^2\) terms: \((-2.3-3.7)xy^2=-6xy^2\). 4. Combine the \(x^2y^2\) terms: \((1.2+0.5)x^2y^2=1.7x^2y^2\). 5. The result is \(2.7x^2y-6xy^2+1.7x^2y^2\).

\(2.7x^2y-6xy^2+1.7x^2y^2\)

- Check every sign carefully when removing the subtraction parentheses. - Group terms with exactly the same variable part. - Combine the coefficients of each group separately.

1. Remove the parentheses, reversing the signs in the polynomial being subtracted: \(7.2u^3v^2-4.1uv^3+2.5w-3.8u^3v^2-1.9uv^3+0.4w+1.5uv^3-3.2w\). 2. Combine the \(u^3v^2\) terms: \((7.2-3.8)u^3v^2=3.4u^3v^2\). 3. Combine the \(uv^3\) terms: \((-4.1-1.9+1.5)uv^3=-4.5uv^3\). 4. Combine the \(w\) terms: \((2.5+0.4-3.2)w=-0.3w\). 5. The result is \(3.4u^3v^2-4.5uv^3-0.3w\).

\(3.4u^3v^2-4.5uv^3-0.3w\)

- Group terms with the same variable part. - Reverse every sign in an expression being subtracted. - Only terms with identical variables and exponents are like terms. - Combine the constant terms separately.

1. Substitute the expressions: \((4x^2-7y+2)+(-2x^2+3y-5)+(x^2-y+1)\). Combine like terms: \((4-2+1)x^2+(-7+3-1)y+(2-5+1)=3x^2-5y-2\). 2. Substitute the expressions: \((4x^2-7y+2)-(-2x^2+3y-5)-(x^2-y+1)\). Remove the subtraction parentheses: \(4x^2-7y+2+2x^2-3y+5-x^2+y-1\). Combine like terms: \((4+2-1)x^2+(-7-3+1)y+(2+5-1)=5x^2-9y+6\).

1) \(3x^2-5y-2\) 2) \(5x^2-9y+6\)

- Group terms with identical variables and exponents. - Reverse every sign in an expression being subtracted. - Simplify the expression inside parentheses before subtracting it.

1. For part a, substitute and combine like terms: \((4.5-2.5+1)x^2+(-5+3-1.5)x+(7-4+2)=3x^2-3.5x+5\). 2. For part b, first find \(B+C=(-2.5+1)x^2+(3-1.5)x+(-4+2)=-1.5x^2+1.5x-2\). 3. Subtract this expression from \(A\): \((4.5x^2-5x+7)-(-1.5x^2+1.5x-2)\). 4. Remove the parentheses and combine like terms: \(4.5x^2+1.5x^2-5x-1.5x+7+2=6x^2-6.5x+9\).

a) \(3x^2-3.5x+5\) b) \(6x^2-6.5x+9\)

- Multiply the monomial by every term in the parentheses. - Pay close attention to signs when the monomial is negative. - Remember that \(x \cdot x = x^2\).

1. \(7x(2x - 3y) = 14x^2 - 21xy\). 2. \(-3a(4b + 2a) = -12ab - 6a^2\). 3. \(0.4z(10z - 5) = 4z^2 - 2z\).

1) \(14x^2 - 21xy\) 2) \(-12ab - 6a^2\) 3) \(4z^2 - 2z\)

- Multiply the factor outside the parentheses by every term inside. - Pay close attention to signs when the outside factor is negative. - Multiply both the numerical coefficients and the variable factors.

1. \(9(3x - 5y) = 27x - 45y\). 2. \(-4(2a + 6b - 3) = -8a - 24b + 12\). 3. \(2m(4n - 7p) = 8mn - 14mp\).

1) \(27x - 45y\) 2) \(-8a - 24b + 12\) 3) \(8mn - 14mp\)

- Multiply the monomial by every term in the polynomial. - Pay close attention to signs when the monomial is negative. - When multiplying powers with the same base, add their exponents.

1. For a), \(5a(2a^2 - 3a + 4) = 10a^3 - 15a^2 + 20a\). 2. For b), \(-2x^2(3x - 5) = -6x^3 + 10x^2\). 3. For c), \((4y^2 - y - 6)(-3y) = -12y^3 + 3y^2 + 18y\).

a) \(10a^3 - 15a^2 + 20a\) b) \(-6x^3 + 10x^2\) c) \(-12y^3 + 3y^2 + 18y\)

- Multiply the outside monomial by every term inside the parentheses. - Pay close attention to signs when multiplying by a negative monomial. - Add exponents when multiplying powers with the same base. - Writing each partial product can prevent errors.

1. \(5x(3x^2 - 2xy + 4y^2) = 15x^3 - 10x^2y + 20xy^2\). 2. \(-4ab(2a - 5b - 1) = -8a^2b + 20ab^2 + 4ab\). 3. \(\frac{1}{2}m(4m^2 + 8mn - 6n^2) = 2m^3 + 4m^2n - 3mn^2\).

1) \(15x^3 - 10x^2y + 20xy^2\) 2) \(-8a^2b + 20ab^2 + 4ab\) 3) \(2m^3 + 4m^2n - 3mn^2\)

- Multiply the outside monomial by every term in the polynomial. - Pay close attention to signs when multiplying by a negative monomial. - Add exponents when multiplying powers with the same base.

1. \(5x(3x^2 - 4x + 2) = 15x^3 - 20x^2 + 10x\). 2. \(-2ab(4a^2 - 3ab + b^2) = -8a^3b + 6a^2b^2 - 2ab^3\).

1) \(15x^3 - 20x^2 + 10x\) 2) \(-8a^3b + 6a^2b^2 - 2ab^3\)

- Multiply the monomial by every term in the polynomial. - Pay close attention to signs when multiplying by a negative number. - Add exponents when multiplying powers with the same base. - Break the work into three separate products.

1. Distribute \(-1.5a\): \(-1.5a(4a^2 - 2a + 0.8) = -6a^3 + 3a^2 - 1.2a\).

\(-6a^3 + 3a^2 - 1.2a\)

- Multiply every term in the first binomial by every term in the second. - Check signs carefully. - Combine terms only when their variable parts and exponents match.

1. \((x + 7)(x - 2) = x^2 - 2x + 7x - 14 = x^2 + 5x - 14\). 2. \((3a - 4)(2a + 5) = 6a^2 + 15a - 8a - 20 = 6a^2 + 7a - 20\). 3. \((5 - 2b)(b + 3) = 5b + 15 - 2b^2 - 6b = -2b^2 - b + 15\).

1) \(x^2 + 5x - 14\) 2) \(6a^2 + 7a - 20\) 3) \(-2b^2 - b + 15\)

- Multiply every term in the first binomial by every term in the second. - Pay close attention to the signs of products involving negative terms. - Combine like terms after expanding.

1. \((x + 8)(x - 2) = x^2 - 2x + 8x - 16 = x^2 + 6x - 16\). 2. \((a - 4)(b + 5) = ab + 5a - 4b - 20\). 3. \((y - 3)(y - 6) = y^2 - 6y - 3y + 18 = y^2 - 9y + 18\).

1) \(x^2 + 6x - 16\) 2) \(ab + 5a - 4b - 20\) 3) \(y^2 - 9y + 18\)

- Multiply every term in one factor by every term in the other. - Apply the sign rules carefully. - Combine terms only when their variables and exponents match exactly.

1. \((4x + 3)(2x - 5) = 8x^2 - 20x + 6x - 15 = 8x^2 - 14x - 15\). 2. \((a - 6b)(2a + b) = 2a^2 + ab - 12ab - 6b^2 = 2a^2 - 11ab - 6b^2\). 3. \((y^2 + 3)(y^2 - 4) = y^4 - 4y^2 + 3y^2 - 12 = y^4 - y^2 - 12\).

1) \(8x^2 - 14x - 15\) 2) \(2a^2 - 11ab - 6b^2\) 3) \(y^4 - y^2 - 12\)

- Expand each product separately. - Subtract the entire second polynomial. - Identify which variable terms cancel and what remains.

1. \((n+3)(n+4)=n^2+7n+12\). 2. \((n+1)(n+6)=n^2+7n+6\). 3. Subtract: \((n^2+7n+12)-(n^2+7n+6)=6\). 4. The \(n^2\) and \(n\) terms cancel completely, so the result is the constant \(6\) for every value of \(n\).

First product: \(n^2+7n+12\) Second product: \(n^2+7n+6\) Difference: \(6\) The value is independent of \(n\) because all variable terms cancel.

- Divide every term in the polynomial by the divisor. - Subtract exponents when dividing powers with the same nonzero base. - Apply the sign rules carefully.

1. Divide each term by \(2.5\): \(15b^3 \div 2.5 = 6b^3\), \(-10b^2 \div 2.5 = -4b^2\), and \(5b \div 2.5 = 2b\). The result is \(6b^3 - 4b^2 + 2b\). 2. Divide each term by \(-4a^2\): \(3a^2\), \(-2a\), and \(1\). The result is \(3a^2 - 2a + 1\).

1) \(6b^3 - 4b^2 + 2b\) 2) \(3a^2 - 2a + 1\)

- Dividing by a fraction is equivalent to multiplying by its reciprocal. - Divide every term by the monomial. - Subtract exponents on the common variable base.

1. Divide each coefficient by \(\frac{6}{5}\) and subtract \(2\) from each exponent of \(z\): \(-10z^4 + 15z^2 - 5\). 2. Divide each coefficient by \(-\frac{1}{4}\) and subtract \(1\) from each exponent of \(x\): \(-3x^2 + 2x - 4\).

1) \(-10z^4 + 15z^2 - 5\) 2) \(-3x^2 + 2x - 4\)

- Divide each term separately. - A variable with no written exponent has exponent \(1\). - Check by multiplying your result by the divisor.

1. Divide each term by \(-6a\): \(-12a^3 \div (-6a) = 2a^2\), \(18a^2 \div (-6a) = -3a\), and \(-6a \div (-6a) = 1\). 2. The result is \(2a^2 - 3a + 1\).

\(2a^2 - 3a + 1\)

- Divide every term in the polynomial by the monomial. - Subtract exponents for each common variable base. - Track the sign of each term.

1. Divide the first term: \(12a^3b^2 \div 3ab^2 = 4a^2\). 2. Divide the second term: \(6a^2b^3 \div 3ab^2 = 2ab\). 3. Divide the third term: \(-3ab^4 \div 3ab^2 = -b^2\). 4. The result is \(4a^2 + 2ab - b^2\).

\(4a^2 + 2ab - b^2\)

- Divide each term by the monomial. - Subtract exponents when dividing powers with the same nonzero base. - Account for the negative divisor in every term.

1. Divide each coefficient by \(-0.5\): \(-2\), \(5\), and \(-1\). 2. Subtract \(2\) from each exponent of \(p\): \(p^3\), \(p^2\), and \(p\). 3. The result is \(-2p^3 + 5p^2 - p\).

\(-2p^3 + 5p^2 - p\)

- Divide each term in the polynomial by the monomial. - Multiply by the reciprocal when dividing fractional coefficients. - Subtract exponents for each common base.

1. \(6x^4y^3 \div \left(\frac{3}{2}x^2y\right) = 4x^2y^2\). 2. \(-\frac{3}{4}x^3y^2 \div \left(\frac{3}{2}x^2y\right) = -\frac{1}{2}xy\). 3. \(\frac{1}{2}x^2y^4 \div \left(\frac{3}{2}x^2y\right) = \frac{1}{3}y^3\). 4. The result is \(4x^2y^2 - \frac{1}{2}xy + \frac{1}{3}y^3\).

\(4x^2y^2 - \frac{1}{2}xy + \frac{1}{3}y^3\)

- Divide every term by the monomial. - Dividing by a fraction is multiplying by its reciprocal. - Cancel common nonzero variable factors.

1. Divide the first term: \(12a^2b \div \left(\frac{6}{5}ab\right) = 10a\). 2. Divide the second term: \(-18ab^2 \div \left(\frac{6}{5}ab\right) = -15b\). 3. Divide the third term: \(6ab \div \left(\frac{6}{5}ab\right) = 5\). 4. The result is \(10a - 15b + 5\).

\(10a - 15b + 5\)

- Group terms with exactly the same variables and exponents. - Include the sign attached to each coefficient. - Opposite terms cancel.

1. Combine the \(x^2\) terms: \(9x^2-12x^2=-3x^2\). 2. Combine the \(xy\) terms: \(-5xy+5xy=0\). 3. Combine the \(y^2\) terms: \(3y^2-2y^2=y^2\). 4. The simplified expression is \(-3x^2+y^2\).

\(-3x^2+y^2\)

- Distinguish between \(a^2b\) and \(ab^2\); they are not like terms. - Express \(\frac{1}{2}\) as a decimal to combine it with \(3.5\). - Keep the constant term in the final expression.

1. The \(a^2b\) terms are like terms. Since \(\frac{1}{2}=0.5\), their coefficients have sum \(3.5+0.5=4\). 2. The \(ab^2\) terms are like terms, and their coefficients have sum \(-2+4=2\). 3. The constant \(-7\) has no like term. 4. The simplified expression is \(4a^2b+2ab^2-7\).

\(4a^2b+2ab^2-7\)

- Identify terms with exactly the same variables and exponents. - Group like terms before adding their coefficients. - Include the sign attached to each coefficient. - Combine only the numerical coefficients.

1. Combine the \(x^2\) terms: \((2.4+0.8)x^2=3.2x^2\). 2. Combine the \(xy\) terms: \((-1.5+0.15)xy=-1.35xy\). 3. The term \(-3.2y^2\) has no like term. 4. The simplified expression is \(3.2x^2-1.35xy-3.2y^2\).

\(3.2x^2-1.35xy-3.2y^2\)

- Compare the exponents carefully; \(a^2b\) and \(ab^2\) are not like terms. - A term with no written coefficient has coefficient \(1\). - Group terms with identical variable parts. - Keep constants separate from variable terms.

1. The \(a^2b\) terms are like terms. Their coefficients have sum \(1+1.2=2.2\). 2. The \(ab^2\) terms are like terms. Their coefficients have sum \(-0.4+0.15=-0.25\). 3. The constant \(-0.75\) has no like term. 4. The simplified expression is \(2.2a^2b-0.25ab^2-0.75\).

\(2.2a^2b-0.25ab^2-0.75\)

- Group terms with exactly the same variable and exponent. - Include the sign attached to each coefficient. - Opposite terms cancel. - Arrange like terms next to one another before combining them.

1. Combine the \(x^3\) terms: \(12x^3-5x^3=7x^3\). 2. Combine the \(x^2\) terms: \(-7x^2+7x^2=0\). 3. The terms \(4x\) and \(-9\) have no like terms. 4. The simplified expression is \(7x^3+4x-9\).

\(7x^3+4x-9\)

- Equivalent expressions have the same value for every allowed value of the variable. - Simplify the more complicated expression first. - Group all \(b^2\) terms, all \(b\) terms, and all constants. - Compare the simplified result with \(T_2\).

1. Combine the \(b^2\) terms in \(T_1\): \(6b^2-2b^2=4b^2\). 2. Combine the \(b\) terms: \(-4b+5b=b\). 3. Combine the constants: \(3-8=-5\). 4. Therefore, \(T_1=4b^2+b-5\), which is identical to \(T_2\). 5. The expressions are equivalent because they have the same value for every value of \(b\).

Yes. Simplifying \(T_1\) gives \(4b^2+b-5\), which is identical to \(T_2\).

- Rewrite each repeated product as a power. - Combine only terms with identical variable parts. - Count how many copies of each like term appear.

1. \(k^2 + k^2 + k^2 = 3k^2\). 2. \(x^3 + y^2 + x^3 + y^2 = 2x^3 + 2y^2\). 3. \(a^2 + a^2 + a^2 = 3a^2\). 4. \(m^3 + n^2 + m^3 = 2m^3 + n^2\).

1) \(3k^2\) 2) \(2x^3 + 2y^2\) 3) \(3a^2\) 4) \(2m^3 + n^2\)

- Group terms with the same variables and exponents. - Adding a negative term is the same as subtracting it. - Combine the coefficients of each group separately. - A term with coefficient \(0\) disappears.

1. Combine the \(x^2\) terms: \((9-5-3)x^2=x^2\). 2. Combine the \(y\) terms: \((-4+4)y=0\). 3. The simplified expression is \(x^2\).

\(x^2\)

- Combine only terms with exactly the same variable part. - Track the signs when adding decimal coefficients. - The terms \(ab\) and \(a^2\) are not like terms. - A group whose coefficients sum to \(0\) disappears.

1. Combine the \(ab\) terms: \((1.8-0.8)ab=ab\). 2. Combine the \(a^2\) terms: \((-0.6-1.4+2)a^2=0\). 3. The simplified expression is \(ab\).

\(ab\)

- Reverse every sign in an expression being subtracted. - Combine only terms with exactly the same variable part. - Distinguish variable terms from terms with powers, such as \(a\) and \(a^2\). - Group like terms before combining their coefficients.

1. For part a, remove the parentheses and combine like terms: \(8x+12y-5x+3y=3x+15y\). 2. For part b, remove the parentheses and combine like terms: \(4a^2-7a-a^2-2a=3a^2-9a\). 3. For part c, remove the parentheses and combine like terms: \(9uv+4v-9uv+6v=10v\). 4. For part d, remove the parentheses and combine like terms: \(-3k+5+7k-2=4k+3\).

a) \(3x+15y\) b) \(3a^2-9a\) c) \(10v\) d) \(4k+3\)

- Match each expression to a sum-square or difference-square identity. - Square every coefficient and variable factor. - Apply the power-of-a-power rule to existing exponents.

1. a) \((5x^3+2y^2)^2=25x^6+20x^3y^2+4y^4\). 2. b) The square terms are \(\frac{1}{9}a^4b^2\) and \(36a^2b^4\). The middle term is \(-2\cdot\left(\frac{1}{3}a^2b\right)\cdot6ab^2=-4a^3b^3\). Therefore, the result is \(\frac{1}{9}a^4b^2-4a^3b^3+36a^2b^4\).

a) \(25x^6+20x^3y^2+4y^4\) b) \(\frac{1}{9}a^4b^2-4a^3b^3+36a^2b^4\)

- Rewrite division as multiplication by a reciprocal if helpful. - Divide every term in the numerator by the divisor. - Track signs carefully when dividing by a negative expression. - Apply the quotient rule for exponents.

1. a) Divide each term by \(4y\): \(12x^2y \div 4y = 3x^2\), \(-8xy \div 4y = -2x\), and \(4y \div 4y = 1\). The result is \(3x^2 - 2x + 1\). 2. b) Divide each term by \(-5ab\): \(-15a^3b^2 \div (-5ab) = 3a^2b\), \(10a^2b \div (-5ab) = -2a\), and \(-5ab \div (-5ab) = 1\). The result is \(3a^2b - 2a + 1\).

a) \(3x^2 - 2x + 1\) b) \(3a^2b - 2a + 1\)

- Check whether every exponent of \(x\) is a nonnegative integer. - Rewrite divisions and radicals involving \(x\) as powers when helpful. - Polynomial coefficients may be fractions or irrational real numbers. - Expand products to reveal standard polynomial form.

1. A polynomial function has real coefficients and only nonnegative integer exponents of \(x\). 2. For a), \(f_1(x)=\frac{1}{5}x^5-2\), so it is a polynomial. 3. For b), the exponent \(\frac{1}{2}\) is not a nonnegative integer, so it is not a polynomial. 4. For c), \((2x+3)^2=4x^2+12x+9\), so it is a polynomial. 5. For d), \(\frac{3}{x}=3x^{-1}\) has a negative exponent, so it is not a polynomial. 6. For e), \(\sqrt{7}\) is a real coefficient and the exponent \(3\) is a nonnegative integer, so it is a polynomial.

a) Yes: \(f_1(x)=\frac{1}{5}x^5-2\) b) No c) Yes: \(f_3(x)=4x^2+12x+9\) d) No e) Yes: \(f_5(x)=\sqrt{7}x^3\)

- Recall what restrictions a polynomial places on coefficients and exponents. - For addition and subtraction, ask whether combining like terms can create a forbidden exponent. - For multiplication, use the exponent rule for multiplying powers with the same base. - Consider why starting with finitely many terms matters.

1. A polynomial has finitely many terms with real coefficients and nonnegative integer exponents. 2. When polynomials are added or subtracted, terms are added or subtracted coefficient by coefficient. Real coefficients remain real, and no new exponent type is created. Combining like terms still leaves finitely many polynomial terms. 3. When polynomial terms are multiplied, their real coefficients are multiplied and their nonnegative integer exponents are added. The product coefficient is real, and the sum of nonnegative integers is still a nonnegative integer. 4. Two finite polynomials produce only finitely many partial products. Combining like terms therefore leaves a polynomial. 5. Thus, polynomials are closed under addition, subtraction, and multiplication.

Polynomials are closed under addition, subtraction, and multiplication. Addition and subtraction only combine coefficients of allowed terms, while multiplication multiplies real coefficients and adds nonnegative integer exponents. Because the starting polynomials have finitely many terms, each result also has finitely many valid polynomial terms.

- Combine the two labeled pieces along each full side. - Find each smaller rectangle's area from the side labels that meet it. - The whole rectangle's area must equal the sum of the four smaller areas.

1. The full horizontal side is \(x+4\), and the full vertical side is \(2x+1\), so the area as a product is \((x+4)(2x+1)\). 2. The four smaller areas are \(2x^2\), \(8x\), \(x\), and \(4\). 3. Their sum is \(2x^2+8x+x+4=2x^2+9x+4\). 4. Therefore, \((x+4)(2x+1)=2x^2+9x+4\).

Product form: \((x+4)(2x+1)\) Sum of regions: \(2x^2+8x+x+4\) Expanded form: \(2x^2+9x+4\)

- Express each rectangle dimension in terms of the square's side length. - Multiply the two rectangle dimensions before comparing areas. - Look for cancellation after setting the area expressions equal.

1. The square's area is \(s^2\). 2. The rectangle has dimensions \(s+4\) and \(s-3\), so its area is \((s+4)(s-3)=s^2+s-12\). 3. Equal areas give \(s^2=s^2+s-12\). 4. Subtract \(s^2\) from both sides: \(s-12=0\), so \(s=12\). 5. The square's side length is \(12\,\text{ft}\).

a) Square area: \(s^2\) Rectangle area: \((s+4)(s-3)=s^2+s-12\) b) \(s^2=s^2+s-12\), so \(s=12\,\text{ft}\)

- Express the original length in terms of the height. - Apply both dimension changes before writing the new area. - Expand the new area before using the stated area increase.

1. The original height is \(x\) and the original length is \(2x\), so the original area is \(2x^2\). 2. The new height is \(x+3\), and the new length is \(2x-2\). 3. The new area is \((x+3)(2x-2)=2x^2+4x-6\). 4. The new area is \(30\,\text{in.}^2\) greater, so \(2x^2+4x-6=2x^2+30\). 5. Simplify: \(4x-6=30\), so \(x=9\). 6. The original dimensions were \(9\,\text{in.}\) by \(18\,\text{in.}\).

a) Original area: \(2x^2\) New area: \((x+3)(2x-2)=2x^2+4x-6\) b) \(2x^2+4x-6=2x^2+30\), so the original dimensions were \(9\,\text{in.}\) by \(18\,\text{in.}\).

- Use common denominators before adding or subtracting fractional coefficients. - With nested grouping symbols, work from the inside out. - Products such as \(ab\) are like terms only with other terms having exactly the same variable factors.

1. For a), remove the parentheses and combine like terms: \(\left(\frac{2}{5} - \frac{1}{10}\right)m + \left(\frac{3}{4} + \frac{1}{2}\right)n = \frac{3}{10}m + \frac{5}{4}n\). 2. For b), expand and combine like terms: \(12ab - 2ab - 10a + ab = 11ab - 10a\). 3. For c), work from the inside out: \(x^2 - (4x - 2x^2 - x) + 7 = x^2 - 3x + 2x^2 + 7 = 3x^2 - 3x + 7\).

a) \(\frac{3}{10}m + \frac{5}{4}n\) b) \(11ab - 10a\) c) \(3x^2 - 3x + 7\)

- Express both original sides using one variable. - Apply the two different increases to the correct sides. - Expand both area products and subtract them before solving.

1. The original sides are \(x\) and \(x+5\), so the original area is \(x(x+5)=x^2+5x\). 2. The new sides are \(x+3\) and \(x+7\), so the new area is \((x+3)(x+7)=x^2+10x+21\). 3. The area increase is \((x^2+10x+21)-(x^2+5x)=5x+21\). 4. Set \(5x+21=66\). Then \(5x=45\), so \(x=9\). 5. The original side lengths were \(9\,\text{in.}\) and \(14\,\text{in.}\).

a) Original area: \(x^2+5x\) New area: \(x^2+10x+21\) b) Area increase: \(5x+21\) \(5x+21=66\), so the original side lengths were \(9\,\text{in.}\) and \(14\,\text{in.}\).

- Products such as \(x^2y\) and \(yx^2\) have the same variable part. - Express fractions and decimals in a common form before combining coefficients. - Distribute carefully in part c, including the negative sign before the second set of parentheses.

1. For part a, use the commutative property to recognize \(yx^2=x^2y\) and \(y^2x=xy^2\). Then combine like terms: \(4x^2y+3x^2y-2xy^2-5xy^2=7x^2y-7xy^2\). 2. For part b, \(ba=ab\). Combine the coefficients: \(\frac{3}{4}-\frac{1}{2}+2.5=0.75-0.5+2.5=2.75\). The result is \(2.75ab\), or \(\frac{11}{4}ab\). 3. For part c, distribute and remove the parentheses: \(z(2z-3)-(z^2+4z)=2z^2-3z-z^2-4z=z^2-7z\).

a) \(7x^2y-7xy^2\) b) \(2.75ab\), or \(\frac{11}{4}ab\) c) \(z^2-7z\)

- Distribute the factor outside parentheses to every term inside. - Pay close attention when a product or set of parentheses is subtracted. - Look for terms that cancel.

1. For a), \(5rs - 10r - 3rs - 12s + 10r = 2rs - 12s\). 2. For b), \(2x^2 - 3xy - x^2 + 2xy = x^2 - xy\). 3. For c), \(k^2m - (k^2m - k^2) + m = k^2m - k^2m + k^2 + m = k^2 + m\).

a) \(2rs - 12s\) b) \(x^2 - xy\) c) \(k^2 + m\)

- Expand each component before combining them. - Apply the factor \(\frac{1}{4}\) to both terms in the product result. - Combine like terms at the end.

1. a) \((k - 0.5)^2 = k^2 - k + 0.25\), and \(k(1 - k) = k - k^2\). Their sum is \(0.25\). 2. b) The square is \(\frac{1}{4}x^2 + 4x + 16\), and \((x - 2)(x + 2) = x^2 - 4\). Therefore, \(\frac{1}{4}x^2 + 4x + 16 - \frac{1}{4}(x^2 - 4) = 4x + 17\).

a) \(0.25\) b) \(4x + 17\)

- Simplify each major product or square separately. - Apply factors outside parentheses to every term. - Look for terms that cancel.

1. a) The difference of the first two squares is \(4ab\). Also, \(-(2a + b)(2a - b) = -(4a^2 - b^2)\). Combining with \(4a^2\) gives \(4ab + b^2\). 2. b) Expand: \(3(x^2 - 4x + 4) - 2(x^2 - 9) - x^2\). Combining like terms gives \(-12x + 30\).

a) \(4ab + b^2\) b) \(-12x + 30\)

- Expand both binomial squares. - Distribute the subtraction sign across the second polynomial. - Identify terms that cancel.

1. \((3a + 2)^2 = 9a^2 + 12a + 4\). 2. \((3a - 2)^2 = 9a^2 - 12a + 4\). 3. Subtracting gives \(9a^2 + 12a + 4 - 9a^2 + 12a - 4 = 24a\).

\(24a\)

- Simplify the conjugate product first. - Expand the binomial square separately. - Apply the subtraction sign before combining like terms.

1. Use the difference of squares: \((x + 5)(x - 5) = x^2 - 25\), so the first part is \(2x^2 - 50\). 2. Expand \((x - 3)^2 = x^2 - 6x + 9\). 3. Subtract and combine: \(2x^2 - 50 - x^2 + 6x - 9 = x^2 + 6x - 59\).

\(x^2 + 6x - 59\)

- Expand the square before applying the outside factor. - Recognize the conjugate product in the second term. - Distribute the subtraction sign carefully.

1. Expand the square: \((2z + 4)^2 = 4z^2 + 16z + 16\). Multiplying by \(\frac{1}{2}\) gives \(2z^2 + 8z + 8\). 2. Use the difference of squares: \((z - 1)(z + 1) = z^2 - 1\). 3. Subtract: \(2z^2 + 8z + 8 - (z^2 - 1) = z^2 + 8z + 9\).

\(z^2 + 8z + 9\)

- \(a^2b\) and \(ab^2\) are different variable parts. - Express fractional and decimal coefficients in a common form. - Keep the constant term in the final expression.

1. Combine the \(a^2b\)-terms: \(\frac{2}{5}-0.6=0.4-0.6=-0.2=-\frac{1}{5}\). 2. Combine the \(ab^2\)-terms: \(\frac{3}{2}-\frac{3}{4}=\frac{6}{4}-\frac{3}{4}=\frac{3}{4}\). 3. Keep the constant term \(5\). The result is \(-\frac{1}{5}a^2b+\frac{3}{4}ab^2+5\).

\(-\frac{1}{5}a^2b+\frac{3}{4}ab^2+5\)

- Simplify Expression A before comparing. - \(a^2\), \(ab\), and \(b\) are different variable parts. - Compare every coefficient and sign after simplifying.

1. Combine the \(a^2\)-terms in Expression A: \(1.5a^2-0.5a^2=a^2\). 2. Combine the \(ab\)-terms: \(-2.4ab+3.1ab=0.7ab\). 3. The \(-0.6b\) term remains unchanged. 4. Expression A simplifies to \(a^2+0.7ab-0.6b\), which is identical to Expression B.

Yes. The expressions are equivalent because Expression A simplifies to \(a^2+0.7ab-0.6b\).

- Combine terms with the same powers of \(x\). - Identify the extra term in the simplified expression. - Add its opposite to make it cancel.

1. Combine the quadratic terms: \(-4x^2-2x^2+7x^2=x^2\). 2. Combine the linear terms: \(9x-12x=-3x\). 3. Therefore, \(T=x^2-3x\). 4. To obtain \(x^2\), add the opposite of \(-3x\), which is \(3x\): \(x^2-3x+3x=x^2\).

a) \(x^2-3x\) b) \(3x\)

- Convert every coefficient to the same numerical form. - \(x^2\)-terms and \(x\)-terms are not like terms. - Combine constants separately.

1. Convert the coefficients: \(\frac{3}{4}=0.75\) and \(1\frac{1}{2}=1.5\). 2. Combine the \(x^2\)-terms: \(1.2x^2-0.8x^2=0.4x^2\). 3. Combine the \(x\)-terms: \(-0.75x+1.5x=0.75x\). 4. Combine the constants: \(0.5-1.1=-0.6\). 5. The simplified expression is \(0.4x^2+0.75x-0.6\).

\(0.4x^2+0.75x-0.6\)

- Subtract the known polynomial from the sum. - Reverse every sign in the polynomial being subtracted. - Combine like terms by degree.

1. Let the missing polynomial be \(Q\): \((2x^2+4x-1)+Q=5x^2-3x+8\). 2. Subtract the known polynomial: \(Q=(5x^2-3x+8)-(2x^2+4x-1)\). 3. Distribute the negative sign: \(Q=5x^2-3x+8-2x^2-4x+1\). 4. Combine like terms: \(Q=3x^2-7x+9\).

\(3x^2-7x+9\)

- Distribute a subtraction sign to every term in the following parentheses. - Combine like terms before solving. - Check whether the quadratic terms in part b) cancel. - Group variable terms and constants separately.

1. For a), remove the parentheses and combine like terms: \(10x - 6 = 24\). Add \(6\): \(10x = 30\). Divide by \(10\): \(x = 3\). 2. For b), distribute the subtraction: \(2y^2 + 9y - 15 - 2y^2 + y - 5 = 60\). 3. The quadratic terms cancel, so \(10y - 20 = 60\). Add \(20\): \(10y = 80\). Divide by \(10\): \(y = 8\). 4. For c), distribute the subtraction and remove the parentheses: \(25 - 8z + 3 + 3z - 2 = 1\). 5. Combine like terms: \(-5z + 26 = 1\). Subtract \(26\): \(-5z = -25\). Divide by \(-5\): \(z = 5\).

a) \(x = 3\) b) \(y = 8\) c) \(z = 5\)

- Put the entire polynomial being subtracted in parentheses. - Reverse every sign in the second polynomial. - Compare corresponding coefficients in the two results.

1. \(P-Q=(12x^2+4x-5)-(5x^2-2x+7)\). 2. Distribute the negative sign and combine like terms: \(12x^2+4x-5-5x^2+2x-7=7x^2+6x-12\). 3. \(Q-P=(5x^2-2x+7)-(12x^2+4x-5)\). 4. Simplify: \(5x^2-2x+7-12x^2-4x+5=-7x^2-6x+12\). 5. The second result is the opposite of the first because \(Q-P=-(P-Q)\).

a) \(7x^2+6x-12\) b) \(-7x^2-6x+12\). It is the opposite of the result in part a.

- Distribute the negative sign to every term in the middle polynomial. - Group terms by degree. - Use common denominators for each group of coefficients.

1. Remove the parentheses: \(2a^3-\frac{5}{8}a^2+\frac{1}{3}a-\frac{3}{4}a^3-a^2+\frac{1}{6}a-1+\frac{1}{2}a^3-\frac{1}{4}\). 2. Combine the cubic terms: \(2-\frac{3}{4}+\frac{1}{2}=\frac{7}{4}\). 3. Combine the quadratic terms: \(-\frac{5}{8}-1=-\frac{13}{8}\). 4. Combine the linear terms: \(\frac{1}{3}+\frac{1}{6}=\frac{1}{2}\). 5. Combine the constants: \(-1-\frac{1}{4}=-\frac{5}{4}\). 6. The simplified polynomial is \(\frac{7}{4}a^3-\frac{13}{8}a^2+\frac{1}{2}a-\frac{5}{4}\).

\(\frac{7}{4}a^3-\frac{13}{8}a^2+\frac{1}{2}a-\frac{5}{4}\)

- Convert all coefficients to a common form. - \(a^2b\) and \(ab^2\) are not like terms. - Remove the parentheses before combining coefficients.

1. Convert the fractional coefficients: \(\frac{1}{2}=0.5\) and \(\frac{1}{4}=0.25\). 2. Remove the parentheses: \(0.5a^2b-0.3ab^2+1.5a^2b-0.7ab^2+0.25a^2b\). 3. Combine the \(a^2b\)-terms: \(0.5+1.5+0.25=2.25\). 4. Combine the \(ab^2\)-terms: \(-0.3-0.7=-1\). 5. The simplified expression is \(2.25a^2b-ab^2\).

\(2.25a^2b-ab^2\)

- Work from the inner parentheses outward. - Simplify the entire bracketed expression before subtracting it. - Reverse every sign when removing the outer brackets.

1. Simplify inside the brackets: \(3y^2-2x^2-5xy+y^2+x^2-5xy\). 2. Combine like terms inside the brackets: \(-x^2-10xy+4y^2\). 3. Subtract the bracketed expression: \(4x^2-[-x^2-10xy+4y^2]=4x^2+x^2+10xy-4y^2\). 4. The result is \(5x^2+10xy-4y^2\).

\(5x^2+10xy-4y^2\)

- Work from the innermost grouping symbols outward. - Pay close attention to each negative sign before a grouped expression. - Treat \(a^2\) terms and \(a\) terms as different kinds of terms. - Combine like terms whenever a grouped expression has been simplified.

1. Simplify the innermost grouping: \(2a-(3a^2-a)=2a-3a^2+a=3a-3a^2\). 2. Substitute into the expression in braces: \(5a^2-[3a-3a^2]+4a\). 3. Remove the brackets and combine like terms: \(5a^2-3a+3a^2+4a=8a^2+a\). 4. Subtract the expression in braces: \(10a^2-(8a^2+a)=2a^2-a\).

\(2a^2-a\)

- Represent the unknown second expression with a variable. - Use the relationship between a difference, its first expression, and its second expression. - Distribute the negative sign carefully before combining like terms.

1. Let the second expression be \(S\). Then \((7a^2-2)-S=3a^2-5a-4\). 2. Solve for \(S\): \(S=(7a^2-2)-(3a^2-5a-4)\). 3. Distribute the negative sign: \(S=7a^2-2-3a^2+5a+4\). 4. Combine like terms: \(S=4a^2+5a+2\). 5. Therefore, \(3a^2-5a-4=(7a^2-2)-(4a^2+5a+2)\).

\((7a^2-2)-(4a^2+5a+2)\)

- Add the two known expressions first. - Subtract their sum from the target polynomial. - Combine only terms with exactly the same variables and exponents.

1. Let the third expression be \(T\). Then \((2a^2b+5)+(-ab^2-3a)+T=a^2b-3ab^2+4a-7\). 2. Combine the known expressions: \(2a^2b-ab^2-3a+5+T=a^2b-3ab^2+4a-7\). 3. Subtract the known sum from the target polynomial: \(T=(a^2b-3ab^2+4a-7)-(2a^2b-ab^2-3a+5)\). 4. Distribute the negative sign: \(T=a^2b-3ab^2+4a-7-2a^2b+ab^2+3a-5\). 5. Combine like terms: \(T=-a^2b-2ab^2+7a-12\).

\(-a^2b-2ab^2+7a-12\)

- Work from the innermost grouping symbols outward. - Pay close attention to the signs when removing parentheses. - Simplify the algebraic expression before substituting the values. - Substitute \(-2\) for \(b\), including its negative sign.

1. Simplify inside the brackets: \(3ab-(a^2+2ab)=3ab-a^2-2ab=ab-a^2\). 2. Simplify the full expression: \(2a^2-(ab-a^2)=2a^2-ab+a^2=3a^2-ab\). 3. Substitute \(a=5\) and \(b=-2\): \(3\cdot5^2-5\cdot(-2)=75+10=85\).

\(85\)

- First simplify using the symbols \(U\) and \(V\). - Track the signs carefully when removing parentheses. - After substituting, group the \(m^2\) terms and the \(n\) terms separately.

1. Simplify \(W\) using \(U\) and \(V\): \(2[U-V+U]-3U=2[2U-V]-3U\). 2. Distribute and combine like terms: \(4U-2V-3U=U-2V\). 3. Substitute the given expressions: \((3m^2-5n)-2(m^2+2n)\). 4. Distribute and combine like terms: \(3m^2-5n-2m^2-4n=m^2-9n\).

\(m^2-9n\)

- Work from the innermost grouping symbols outward. - Reverse every sign when subtracting a grouped expression. - Simplify the algebraic expression before substituting the values. - Remember that the square of a negative number is positive.

1. Remove the inner parentheses: \(2a^2b-[3ab^2-4a^2b+2ab^2+5a^2b]\). 2. Combine like terms inside the brackets: \(2a^2b-[5ab^2+a^2b]\). 3. Remove the brackets and combine like terms: \(2a^2b-5ab^2-a^2b=a^2b-5ab^2\). 4. Substitute \(a=-2\) and \(b=3\): \((-2)^2\cdot3-5\cdot(-2)\cdot3^2\). 5. Evaluate: \(4\cdot3-5\cdot(-2)\cdot9=12+90=102\).

\(102\)

- Work from the innermost grouping symbols outward. - Group like terms, such as all \(x^2\) terms, before combining them. - The product of two negative numbers is positive. - Check every sign after removing a grouped expression.

1. Remove the innermost parentheses: \(4xy-\{2x^2-[3xy-x^2+2xy+4x^2]-5xy\}\). 2. Combine like terms inside the brackets: \(4xy-\{2x^2-[5xy+3x^2]-5xy\}\). 3. Remove the brackets: \(4xy-\{2x^2-5xy-3x^2-5xy\}\). 4. Combine like terms inside the braces: \(4xy-\{-x^2-10xy\}\). 5. Remove the braces and combine like terms: \(4xy+x^2+10xy=x^2+14xy\). 6. Substitute \(x=-1\) and \(y=-4\): \((-1)^2+14\cdot(-1)\cdot(-4)=1+56=57\).

\(57\)

- Work from the innermost grouping symbols outward. - Reverse every sign when subtracting a grouped expression. - Simplify completely before substituting the values. - Evaluate exponents before multiplication.

1. Remove the innermost parentheses: \(x^2y-\{2xy^2-[3x^2y-xy^2-x^2y]\}\). 2. Combine like terms inside the brackets: \(x^2y-\{2xy^2-[2x^2y-xy^2]\}\). 3. Remove the brackets: \(x^2y-\{2xy^2-2x^2y+xy^2\}\). 4. Combine like terms inside the braces: \(x^2y-\{3xy^2-2x^2y\}\). 5. Remove the braces and combine like terms: \(x^2y-3xy^2+2x^2y=3x^2y-3xy^2\). 6. Substitute \(x=-1\) and \(y=\frac{2}{3}\): \(3\cdot(-1)^2\cdot\frac{2}{3}-3\cdot(-1)\cdot\left(\frac{2}{3}\right)^2\). 7. Evaluate: \(2+\frac{4}{3}=\frac{10}{3}\).

\(\frac{10}{3}\), or \(3\frac{1}{3}\)

- Work backward from a product term to its missing factor. - Division is the inverse of multiplication. - Check both the sign and the exponent of each missing term.

1. For a), divide the first product term by \(3x\): \(12x^2 \div 3x = 4x\). Checking gives \(4x(-4y) = -16xy\). 2. For b), divide the second product term by \(-5b\): \(-15b^2 \div (-5b) = 3b\). 3. For c), divide \(x^2\) by \(x\): \(x^2 \div x = x\).

a) \(4x\) b) \(3b\) c) \(x\)

- Determine the sign of each product before simplifying it. - Fractions and decimals follow the same multiplication rules as integers. - When multiplying powers with the same base, add the exponents. A variable with no written exponent has exponent \(1\).

1. For a), \(\frac{2}{3}z(6z^2 - 9z + 12) = 4z^3 - 6z^2 + 8z\). 2. For b), \(-1.5m(2m^2 - 4m + 6) = -3m^3 + 6m^2 - 9m\). 3. For c), \((10x^3 - 5x^2 + 25x)\left(-\frac{1}{5}x\right) = -2x^4 + x^3 - 5x^2\).

a) \(4z^3 - 6z^2 + 8z\) b) \(-3m^3 + 6m^2 - 9m\) c) \(-2x^4 + x^3 - 5x^2\)

- Divide each product term by \(4a^2\) to recover the corresponding term in the polynomial. - Division reverses multiplication. - For part b), distribute \(-2ab\) to every term in the polynomial. - Check signs and exponents carefully.

1. Divide each term by \(4a^2\): \(12a^3 \div 4a^2 = 3a\), \(-8a^2b \div 4a^2 = -2b\), and \(20a^2 \div 4a^2 = 5\). The original polynomial is \(3a - 2b + 5\). 2. Multiply by \(-2ab\): \(-2ab(3a - 2b + 5) = -6a^2b + 4ab^2 - 10ab\).

a) \(3a - 2b + 5\) b) \(-6a^2b + 4ab^2 - 10ab\)

- Multiply the outside factor by every term inside the parentheses. - Check the sign of each product, especially when two negative factors are multiplied. - Fractions and decimals distribute in the same way as integers.

1. \(6(3a - 4b + 7) = 18a - 24b + 42\). 2. \((x^2 - 2x - 5)(-3) = -3x^2 + 6x + 15\). 3. \(0.4(10m - 5n + 15) = 4m - 2n + 6\). 4. \((9r + 6s - 3)\left(-\frac{2}{3}\right) = -6r - 4s + 2\).

1) \(18a - 24b + 42\) 2) \(-3x^2 + 6x + 15\) 3) \(4m - 2n + 6\) 4) \(-6r - 4s + 2\)

- Expand the expression yourself before comparing the answers. - Was every term inside the parentheses multiplied by the outside monomial? - What is the product of a quantity and \(1\)?

1. Distribute \(-3x^2\) to every term: \(-3x^2(2x - 4y + 1) = -6x^3 + 12x^2y - 3x^2\). 2. Sarah is correct. 3. Lucas changed \(+1\) to \(-1\) instead of multiplying it by \(-3x^2\).

Sarah is correct. Lucas failed to multiply the final term \(1\) by \(-3x^2\).

- Expand one product at a time. - Pay close attention to the negative factor before the second product. - Combine terms only when they have the same variable factors. - Check that every product was expanded completely.

1. Expand the products: \(5x^2 - 10xy - 12xy + 3y^2 + 22xy\). 2. Combine the \(xy\)-terms: \((-10 - 12 + 22)xy = 0\). 3. The simplified expression is \(5x^2 + 3y^2\).

\(5x^2 + 3y^2\)

- Distribute the negative factor carefully. - Remember that \(km\) and \(mk\) are the same product. - Look for terms that cancel. - Multiply the decimal coefficient just as you would any other number.

1. Expand the first product: \(0.5k(4k - 6m) = 2k^2 - 3km\). 2. Expand the second product: \(-2m(k - 3m) = -2km + 6m^2\). 3. Combine all terms: \(2k^2 - 3km - 2km + 6m^2 - 2k^2 = 6m^2 - 5km\).

\(6m^2 - 5km\)

- Multiply every term in the polynomial by the monomial outside. - Multiply the coefficients first, then handle each variable. - When multiplying \(x^2\) by \(x\), add the exponents. - Choose either fraction or decimal form for the coefficient calculation and use it consistently.

1. Multiply the first term: \(5xy\left(\frac{2}{5}x^2y\right) = 2x^3y^2\). 2. Multiply the second term: \(5xy(-10xy^2) = -50x^2y^3\). 3. Multiply the third term: \(5xy\left(\frac{1}{2}y\right) = \frac{5}{2}xy^2\). 4. The product is \(2x^3y^2 - 50x^2y^3 + \frac{5}{2}xy^2\).

\(2x^3y^2 - 50x^2y^3 + \frac{5}{2}xy^2\)

- Pay close attention to the sign of each outside factor. - Group terms by their variable factors before combining. - Terms can be combined only when their variables and exponents match exactly.

1. Expand both products: \(12x^2 - 8xy - 2xy - 10x^2 + 7xy\). 2. Combine the \(x^2\)-terms: \(12x^2 - 10x^2 = 2x^2\). 3. Combine the \(xy\)-terms: \(-8xy - 2xy + 7xy = -3xy\). 4. The result is \(2x^2 - 3xy\).

\(2x^2 - 3xy\)

- Distribute negative factors carefully. - Expand each product separately before combining terms. - Group terms with the same variable factors.

1. Expand each product: \(8ab - 6a^2 - 5a^2 + 10ab - 3a^2 - ab\). 2. Combine the \(a^2\)-terms: \(-6a^2 - 5a^2 - 3a^2 = -14a^2\). 3. Combine the \(ab\)-terms: \(8ab + 10ab - ab = 17ab\). 4. The result is \(-14a^2 + 17ab\).

\(-14a^2 + 17ab\)

- Multiply every term inside each set of parentheses by its outside factor. - Check the sign of each product. - Group the \(a^2\)-terms, \(ab\)-terms, and \(b^2\)-terms separately. - Remember that \(ab = ba\).

1. Expand the first product: \(1.2a(2a - 0.5b) = 2.4a^2 - 0.6ab\). 2. Expand the second product: \(-0.4(3a^2 - 5ab + 2b^2) = -1.2a^2 + 2ab - 0.8b^2\). 3. Expand the third product: \(0.5b(4a - 6b) = 2ab - 3b^2\). 4. Combine like terms: \(1.2a^2 + 3.4ab - 3.8b^2\).

\(1.2a^2 + 3.4ab - 3.8b^2\)

- Distribute each decimal coefficient to every term. - Pay close attention to the minus sign before the second product. - Do not combine \(x^2\)-terms with \(x\)-terms. - Remove all parentheses before grouping like terms.

1. Expand the first product: \(1.5(4x^2 - 2x + 6) = 6x^2 - 3x + 9\). 2. Expand the second product and subtract it: \(-1.2(5x^2 - 10x - 5) = -6x^2 + 12x + 6\). 3. Combine like terms: \(6x^2 - 3x + 9 - 6x^2 + 12x + 6 = 9x + 15\).

\(9x + 15\)

- Express the original length in terms of \(w\). - Expand the new area before subtracting. - Simplify the difference of the two area polynomials. - Compare the algebraic difference with the dimensions of the removed strip.

1. The original length is \(3w\), so the original area is \(3w^2\). 2. The shortened length is \(3w-2\), so the new area is \(w(3w-2)=3w^2-2w\). 3. The area decrease is \(3w^2-(3w^2-2w)=2w\). 4. Set \(2w=12\). Then \(w=6\), so the original length is \(18\,\text{in.}\). 5. Geometrically, shortening the length by \(2\,\text{in.}\) removes a \(2\,\text{in.}\)-by-\(w\) strip, whose area is \(2w\).

a) Original area: \(3w^2\); shortened area: \(3w^2-2w\); area decrease: \(2w\) b) Width: \(6\,\text{in.}\); length: \(18\,\text{in.}\) c) The removed strip has area \(2w\), matching the simplified polynomial difference.

- Write one product for each new rectangular area. - Expand both products before comparing them. - Subtract corresponding polynomial terms and look for cancellation. - Set the simplified difference equal to \(15\).

1. Gardener A's new area is \(a(a+2)=a^2+2a\). 2. Gardener B's new area is \(a(a+5)=a^2+5a\). 3. Subtract the expanded areas: \((a^2+5a)-(a^2+2a)=3a\). 4. Since the difference is \(15\,\text{ft}^2\), \(3a=15\), so \(a=5\).

The original side length is \(5\,\text{ft}\).

- Remember that \(ab = ba\) when identifying like terms. - In part 4, expand both products before subtracting. - Look for terms that cancel in part 3.

1. \((4a + b)(a - 3b) = 4a^2 - 12ab + ab - 3b^2 = 4a^2 - 11ab - 3b^2\). 2. \((2x - 3y)(5x - 2y) = 10x^2 - 4xy - 15xy + 6y^2 = 10x^2 - 19xy + 6y^2\). 3. \((m + n)(m - n) = m^2 - mn + mn - n^2 = m^2 - n^2\). 4. \((x + 2)(x + 3) - x(x + 5) = x^2 + 5x + 6 - x^2 - 5x = 6\).

1) \(4a^2 - 11ab - 3b^2\) 2) \(10x^2 - 19xy + 6y^2\) 3) \(m^2 - n^2\) 4) \(6\)

- Multiply every term in the first factor by every term in the second. - Check the signs of all partial products. - After expanding, look for pairs of terms that cancel.

1. For a), \((y + 4)(y^2 - 4y + 16) = y^3 - 4y^2 + 16y + 4y^2 - 16y + 64 = y^3 + 64\). 2. For b), \((z - 2)(z^2 + 2z + 4) = z^3 + 2z^2 + 4z - 2z^2 - 4z - 8 = z^3 - 8\).

a) \(y^3 + 64\) b) \(z^3 - 8\)

- Expand the product completely before considering the final subtraction. - Remember that \(x \cdot x^2 = x^3\). - Look for terms with equal magnitude and opposite signs.

1. Expand the product: \((2x + 1)(4x^2 - 2x + 1) = 8x^3 - 4x^2 + 2x + 4x^2 - 2x + 1\). 2. Combine terms in the product: \(8x^3 + 1\). 3. Subtract \(8x^3\): \(8x^3 + 1 - 8x^3 = 1\).

\(1\)

- Multiplying a variable by itself produces a square. - Recognize \(ab\) and \(ba\) as like terms. - In descending order, write the highest exponent first. - Recheck every sign.

1. \((2x + 1)(x - 4) = 2x^2 - 8x + x - 4 = 2x^2 - 7x - 4\). 2. \((3a - b)(2a + b) = 6a^2 + 3ab - 2ab - b^2 = 6a^2 + ab - b^2\). 3. \((5 - k)(k + 2) = 5k + 10 - k^2 - 2k = -k^2 + 3k + 10\).

1) \(2x^2 - 7x - 4\) 2) \(6a^2 + ab - b^2\) 3) \(-k^2 + 3k + 10\)

- Add exponents when multiplying powers with the same base. - Track every negative sign through each partial product. - Like terms must match in every variable and exponent.

1. \((2x^2 - 3y^2)(4x^2 + y^2) = 8x^4 + 2x^2y^2 - 12x^2y^2 - 3y^4 = 8x^4 - 10x^2y^2 - 3y^4\). 2. \((3a^2b + 2a)(5ab^2 - 4b) = 15a^3b^3 - 12a^2b^2 + 10a^2b^2 - 8ab = 15a^3b^3 - 2a^2b^2 - 8ab\). 3. \((-x^2y - 2)(3x^2y + 5) = -3x^4y^2 - 5x^2y - 6x^2y - 10 = -3x^4y^2 - 11x^2y - 10\).

1) \(8x^4 - 10x^2y^2 - 3y^4\) 2) \(15a^3b^3 - 2a^2b^2 - 8ab\) 3) \(-3x^4y^2 - 11x^2y - 10\)

- Multiply the first term of the left factor by the entire second factor, then do the same with the second term. - Use the power rules, such as \(a \cdot a^2 = a^3\). - Pay close attention when multiplying by \(-2b\). - Group matching variable terms to find cancellations.

1. Multiply \(a\) by every term: \(a^3 + 2a^2b + 4ab^2 - a\). 2. Multiply \(-2b\) by every term: \(-2a^2b - 4ab^2 - 8b^3 + 2b\). 3. Combine like terms. The \(a^2b\)-terms and \(ab^2\)-terms cancel, leaving \(a^3 - 8b^3 - a + 2b\).

\(a^3 - 8b^3 - a + 2b\)

- There should be six partial products before combining terms. - Check each sign carefully. - Combine only terms with exactly the same variables and exponents. - Mark each term after using it to avoid omissions.

1. Form all six partial products: \(x^3 + 2x^2y - 4x^2y - 8xy^2 + 3xy^2 + 6y^3\). 2. Combine like terms: \(x^3 + (2 - 4)x^2y + (-8 + 3)xy^2 + 6y^3\). 3. The result is \(x^3 - 2x^2y - 5xy^2 + 6y^3\).

\(x^3 - 2x^2y - 5xy^2 + 6y^3\)

- Remember that a negative times a negative is positive. - Arrange like terms next to each other before combining. - Check that every term in one factor was multiplied by every term in the other.

1. Form all partial products: \(6a^3 + 2a^2b - 4ab^2 - 9a^2b - 3ab^2 + 6b^3\). 2. Combine like terms: \(6a^3 + (2 - 9)a^2b + (-4 - 3)ab^2 + 6b^3\). 3. The result is \(6a^3 - 7a^2b - 7ab^2 + 6b^3\).

\(6a^3 - 7a^2b - 7ab^2 + 6b^3\)

- Use the distributive property to multiply every term in one factor by every term in the other. - Apply the sign rules carefully. - Combine terms only when they have the same power of \(x\). - Organizing the partial products in rows may help.

1. Form the partial products: \(8x^3 + 4x^2 - 20x - 6x^2 - 3x + 15\). 2. Combine terms with the same power of \(x\): \(8x^3 - 2x^2 - 23x + 15\).

\(8x^3 - 2x^2 - 23x + 15\)

- Add exponents when multiplying powers with the same base. - Multiply every term in the first polynomial by both terms in the second. - Do not combine terms such as \(a^3b\) and \(a^2b^2\). - Verify that you formed all eight partial products.

1. Form all eight partial products: \(2a^4 + 4a^3b - 3a^3b - 6a^2b^2 + a^2b^2 + 2ab^3 - ab^3 - 2b^4\). 2. Combine like terms: \(2a^4 + a^3b - 5a^2b^2 + ab^3 - 2b^4\).

\(2a^4 + a^3b - 5a^2b^2 + ab^3 - 2b^4\)

- A four-term polynomial times a two-term polynomial creates eight partial products. - Pay close attention when multiplying by \(-3\). - Add exponents when multiplying powers of \(y\). - Sort the partial products by exponent before combining.

1. Form the eight partial products: \(2y^5 - 6y^3 - y^4 + 3y^2 + 4y^3 - 12y - y^2 + 3\). 2. Arrange by descending powers and combine like terms: \(2y^5 - y^4 - 2y^3 + 2y^2 - 12y + 3\).

\(2y^5 - y^4 - 2y^3 + 2y^2 - 12y + 3\)

- Multiply each term in the first polynomial by the entire second polynomial. - Arrange partial products by powers of \(x\) before combining. - Remember that \(x^2 \cdot x^3 = x^5\). - Track the signs carefully, especially in the products involving \(-2x\).

1. Multiply each term in the first factor by the second polynomial: \(x^2(x^3 + 3x^2 - 5x + 2) = x^5 + 3x^4 - 5x^3 + 2x^2\), \(-2x(x^3 + 3x^2 - 5x + 2) = -2x^4 - 6x^3 + 10x^2 - 4x\), and \(x^3 + 3x^2 - 5x + 2\). 2. Combine equal powers: \(x^5 + x^4 - 10x^3 + 15x^2 - 9x + 2\).

\(x^5 + x^4 - 10x^3 + 15x^2 - 9x + 2\)

- Multiply every term in the first factor by every term in the second. - Track the signs carefully. - Use a common denominator to combine fractional coefficients. - Combine only terms with matching variable factors.

1. Form the six partial products: \(\frac{1}{2}a^2 + \frac{2}{9}ab - \frac{4}{3}a - \frac{3}{8}ab - \frac{1}{6}b^2 + b\). 2. Combine the \(ab\)-terms: \(\frac{2}{9}ab - \frac{3}{8}ab = \left(\frac{16}{72} - \frac{27}{72}\right)ab = -\frac{11}{72}ab\). 3. The result is \(\frac{1}{2}a^2 - \frac{11}{72}ab - \frac{4}{3}a - \frac{1}{6}b^2 + b\).

\(\frac{1}{2}a^2 - \frac{11}{72}ab - \frac{4}{3}a - \frac{1}{6}b^2 + b\)

- Multiply every term in the first trinomial by every term in the second. - Apply the sign rules carefully. - Look for pairs of opposite terms after expanding. - Add fractions with the same denominator directly.

1. Form the nine partial products: \(\frac{1}{4}x^2 - \frac{1}{6}xy + \frac{1}{8}xz + \frac{1}{6}xy - \frac{1}{9}y^2 + \frac{1}{12}yz - \frac{1}{8}xz + \frac{1}{12}yz - \frac{1}{16}z^2\). 2. The \(xy\)-terms and \(xz\)-terms cancel. The \(yz\)-terms combine to \(\frac{1}{6}yz\). 3. The result is \(\frac{1}{4}x^2 - \frac{1}{9}y^2 + \frac{1}{6}yz - \frac{1}{16}z^2\).

\(\frac{1}{4}x^2 - \frac{1}{9}y^2 + \frac{1}{6}yz - \frac{1}{16}z^2\)

- Simplifying first is more efficient than substituting immediately. - A minus sign before parentheses changes every sign inside. - Expand each product systematically. - Combine terms with the same power of \(x\).

1. Expand both products: \((2x^2 + 5x - 12) - (2x^2 - 3x - 2)\). 2. Remove the second parentheses and combine like terms: \(2x^2 + 5x - 12 - 2x^2 + 3x + 2 = 8x - 10\). 3. Substitute \(x = 0.75\): \(8 \cdot 0.75 - 10 = 6 - 10 = -4\).

Simplified expression: \(8x - 10\) Value at \(x = 0.75\): \(-4\)

- Break the expression into separate products and simplify each one. - One product is a difference of squares. - Pay close attention to the minus sign before the second product. - Convert the mixed number to an improper fraction before evaluating.

1. Expand the products: \((3n - 2)(n + 3) = 3n^2 + 7n - 6\), \((n - 1)(3n + 2) = 3n^2 - n - 2\), and \((n + 2)(n - 2) = n^2 - 4\). 2. Substitute these expressions and combine like terms: \((3n^2 + 7n - 6) - (3n^2 - n - 2) + (n^2 - 4) - n^2 = 8n - 8\). 3. Convert \(-1\frac{1}{8}\) to \(-\frac{9}{8}\), then evaluate: \(8 \cdot \left(-\frac{9}{8}\right) - 8 = -17\).

Simplified expression: \(8n - 8\) Value at \(n = -1\frac{1}{8}\): \(-17\)

- In a product of two binomials, multiply every term in one factor by every term in the other. - In a chain of multiplication, factors may be regrouped and reordered. - Combine only terms with matching variables and exponents. - Track all signs carefully.

1. \((a + 6)(3a - 2) = 3a^2 - 2a + 18a - 12 = 3a^2 + 16a - 12\). 2. Use the associative property: \(2 \cdot 3(2z - 5) = 6(2z - 5) = 12z - 30\). Then \(4z + 12z - 30 = 16z - 30\).

1) \(3a^2 + 16a - 12\) 2) \(16z - 30\)

- First arrange each polynomial in descending powers. - Track the signs of all partial products. - Multiply every term in one factor by every term in the other. - Combine terms with the same power of \(x\).

1. Rewrite the factors as \((3x^2 - 4x + 5)(x^2 + 2x - 3)\). 2. Form the partial products: \(3x^4 + 6x^3 - 9x^2 - 4x^3 - 8x^2 + 12x + 5x^2 + 10x - 15\). 3. Combine like powers: \(3x^4 + 2x^3 - 12x^2 + 22x - 15\).

\(3x^4 + 2x^3 - 12x^2 + 22x - 15\)

- Arrange the first polynomial by powers of \(a\). - Terms such as \(a^2b\) and \(ab^2\) are not like terms. - Track negative signs carefully. - Combine only terms with exactly matching variable factors.

1. Rewrite the first factor as \(a^2 - 3ab + 2b^2\). 2. Form the partial products: \(4a^3 - 5a^2b - 12a^2b + 15ab^2 + 8ab^2 - 10b^3\). 3. Combine like terms: \(4a^3 - 17a^2b + 23ab^2 - 10b^3\).

\(4a^3 - 17a^2b + 23ab^2 - 10b^3\)

- Multiply every term in one factor by every term in the other. - Apply the sign rules carefully. - Arrange the result by powers before combining. - Add exponents when multiplying powers with the same base.

1. Form the partial products and combine like terms: \(8a^3 - 10a^2b - 12a^2b + 15ab^2 + 4ab^2 - 5b^3 = 8a^3 - 22a^2b + 19ab^2 - 5b^3\). 2. Expand and combine equal powers: \(2x^4 + x^3 - 4x^2 - 6x^3 - 3x^2 + 12x + 4x^2 + 2x - 8 = 2x^4 - 5x^3 - 3x^2 + 14x - 8\).

1) \(8a^3 - 22a^2b + 19ab^2 - 5b^3\) 2) \(2x^4 - 5x^3 - 3x^2 + 14x - 8\)

- A coefficient is the numerical factor of a variable term. - For part c), you may substitute into either the original product or the expanded expression. - Include the sign when combining the \(x^2\)-terms.

1. Expand: \(6x^3 - 15x^2 - 4x^2 + 10x + 2x - 5\). 2. Combine like terms: \(T = 6x^3 - 19x^2 + 12x - 5\). 3. The coefficient of \(x^2\) is \(-19\). 4. For \(x = 2\), \(T = 6 \cdot 2^3 - 19 \cdot 2^2 + 12 \cdot 2 - 5 = -9\).

a) \(6x^3 - 19x^2 + 12x - 5\) b) \(-19\) c) \(-9\)

- A minus sign before parentheses changes every sign inside. - In part 3, factoring out \(m\) may simplify the setup. - When multiplying two binomials, multiply every term in one by every term in the other.

1. \(3(m + 2n) + 4(m - n) = 7m + 2n\). 2. \(2(m + 2n) - (m - n) = m + 5n\). 3. \(m(m + 2n) - m(m - n) = m^2 + 2mn - m^2 + mn = 3mn\). 4. \((m + 2n)(m - n) = m^2 - mn + 2mn - 2n^2 = m^2 + mn - 2n^2\).

1) \(7m + 2n\) 2) \(m + 5n\) 3) \(3mn\) 4) \(m^2 + mn - 2n^2\)

- Simplify the conjugate products before substituting. - Combine like terms. - Use parentheses when squaring the decimal input.

1. Use the difference of squares: \((a - 5)(a + 5) = a^2 - 25\) and \((a - 1)(a + 1) = a^2 - 1\). 2. Thus, \(T(a) = 2a^2 - 26\). 3. Substitute \(a = 0.4\): \(2 \cdot (0.4)^2 - 26 = 2 \cdot 0.16 - 26 = -25.68\).

\(-25.68\)

- Express the original length using the width. - Check why the two changed dimensions are equal. - Expand both areas and subtract before using the given difference.

1. The original rectangle has dimensions \(x\) and \(x+7\), so its area is \(x(x+7)=x^2+7x\). 2. After the changes, both dimensions are \(x+5\), so the square's area is \((x+5)^2=x^2+10x+25\). 3. The difference is \((x^2+10x+25)-(x^2+7x)=3x+25\). 4. Set \(3x+25=55\). Then \(x=10\). 5. The square's side length is \(x+5=15\,\text{in.}\).

a) Original area: \(x^2+7x\) Square area: \(x^2+10x+25\) b) \(3x+25=55\), so the square's side length is \(15\,\text{in.}\).

- Express both dimensions using the original width. - Apply the changes before multiplying the new dimensions. - Use the equal-area condition only after both polynomials are written.

1. The original dimensions are \(b\) and \(b+8\), so the original area is \(b(b+8)=b^2+8b\). 2. The new dimensions are \(b+2\) and \(b+5\), so the new area is \((b+2)(b+5)=b^2+7b+10\). 3. Equal areas give \(b^2+8b=b^2+7b+10\). 4. Simplify: \(b=10\). 5. The original dimensions are \(10\,\text{in.}\) and \(18\,\text{in.}\), so the perimeter is \(2(10+18)=56\,\text{in.}\).

a) Original area: \(b^2+8b\) New area: \(b^2+7b+10\) b) \(b^2+8b=b^2+7b+10\), so the original perimeter is \(56\,\text{in.}\).

- Expand both products before comparing them. - Distribute the subtraction across the entire second product. - Notice what happens to the \(x^2\) terms. - Use the simplified expression, not trial-and-error testing, in part b.

1. Expand the first product: \((x+4)(x+6)=x^2+10x+24\). 2. Expand the second product: \(x(x+2)=x^2+2x\). 3. Subtract: \(D(x)=x^2+10x+24-(x^2+2x)=8x+24\). 4. For part b, set \(8x+24=272\). 5. Then \(8x=248\), so \(x=31\). 6. The four consecutive odd integers are \(31\), \(33\), \(35\), and \(37\).

a) \(D(x)=8x+24\) b) \(31, 33, 35, 37\)

- Express the length in terms of the width. - Apply both dimension changes before expanding. - Compare the two expanded area expressions.

1. The original dimensions are \(x\) and \(3x\), so the original area is \(3x^2\). 2. The new dimensions are \(x+2\) and \(3x-3\), so the new area is \((x+2)(3x-3)=3x^2+3x-6\). 3. The increase is \(3x-6\). 4. Set \(3x-6=15\), giving \(x=7\). 5. The original dimensions were \(7\,\text{in.}\) by \(21\,\text{in.}\).

a) Original area: \(3x^2\) New area: \(3x^2+3x-6\) b) Area increase: \(3x-6\) \(3x-6=15\), so the original dimensions were \(7\,\text{in.}\) by \(21\,\text{in.}\).

- Pay close attention to the minus sign before the binomial product. - Expand the product completely before applying the outside subtraction. - Group terms with the same variable factors.

1. Expand the first product: \(4a(2a - 3b) = 8a^2 - 12ab\). 2. Expand the binomial product: \((a - b)(3a + 2b) = 3a^2 - ab - 2b^2\). 3. Subtract this product and combine like terms: \(8a^2 - 12ab - 3a^2 + ab + 2b^2 - 5a^2 = 2b^2 - 11ab\).

\(2b^2 - 11ab\)

- Express the original length using the width. - Expand the new area after both dimensions change. - Subtract corresponding polynomial terms before solving.

1. The original dimensions are \(x\) and \(x+5\), so the original area is \(x^2+5x\). 2. The new dimensions are \(x+2\) and \(x+8\), so the new area is \((x+2)(x+8)=x^2+10x+16\). 3. The increase is \(5x+16\). 4. Set \(5x+16=41\), giving \(x=5\). 5. The original dimensions were \(5\,\text{ft}\) by \(10\,\text{ft}\).

a) Original area: \(x^2+5x\) New area: \(x^2+10x+16\) b) Area increase: \(5x+16\) \(5x+16=41\), so the original dimensions were \(5\,\text{ft}\) by \(10\,\text{ft}\).

- Expand the redesigned rectangle before comparing areas. - The quadratic terms cancel in the area difference. - For equal opposite changes, compare \((s+d)(s-d)\) with \(s^2\).

1. The original area is \(s^2\). 2. The redesigned area is \((s+6)(s-4)=s^2+2s-24\). 3. The increase is \(2s-24\). Set \(2s-24=16\), so \(s=20\). 4. For equal changes of \(5\,\text{ft}\), the new area is \((s+5)(s-5)=s^2-25\). 5. At \(s=20\), the new area is \(400-25=375\,\text{ft}^2\), not \(400\,\text{ft}^2\). The claim is false.

a) Original area: \(s^2\) Redesigned area: \(s^2+2s-24\) Area increase: \(2s-24\) \(s=20\,\text{ft}\) b) \((s+5)(s-5)=s^2-25\). For \(s=20\), the area is \(375\,\text{ft}^2\), so the claim is false.

- Square coefficients and variable powers separately. - Multiply exponents when raising a power to a power. - Include the factor \(2\) in the middle term.

1. a) The square terms are \(\frac{9}{16}a^4\) and \(\frac{1}{9}b^2\), and the middle term is \(2\left(\frac{3}{4}a^2\right)\left(\frac{1}{3}b\right) = \frac{1}{2}a^2b\). 2. b) The square terms are \(2.25z^6\) and \(4w^4\), and the middle term is \(-2(1.5z^3)(2w^2) = -6z^3w^2\).

a) \(\frac{9}{16}a^4 + \frac{1}{2}a^2b + \frac{1}{9}b^2\) b) \(2.25z^6 - 6z^3w^2 + 4w^4\)

- Consider what happens when an entire negative expression is squared. - Identify the two terms in the binomial. - Square coefficients and variable powers separately. - Remember the factor \(2\) in the middle term.

1. Squaring a negative expression gives the same result as squaring its opposite: \((-1.5a^2b - 0.4ab^2)^2 = (1.5a^2b + 0.4ab^2)^2\). 2. Apply \((A+B)^2=A^2+2AB+B^2\). 3. The square terms are \((1.5a^2b)^2=2.25a^4b^2\) and \((0.4ab^2)^2=0.16a^2b^4\). 4. The middle term is \(2(1.5a^2b)(0.4ab^2)=1.2a^3b^3\). 5. Therefore, the simplified expression is \(2.25a^4b^2+1.2a^3b^3+0.16a^2b^4\).

\(2.25a^4b^2+1.2a^3b^3+0.16a^2b^4\)

- Expand each binomial square separately. - Distribute the subtraction sign across every term in the second polynomial. - Look for terms that cancel.

1. Expand the first square: \((2x^2+0.5y)^2=4x^4+2x^2y+0.25y^2\). 2. Expand the second square: \((2x^2-0.5y)^2=4x^4-2x^2y+0.25y^2\). 3. Subtract the second polynomial: \(4x^4+2x^2y+0.25y^2-4x^4+2x^2y-0.25y^2\). 4. Combine like terms to get \(4x^2y\).

\(4x^2y\)

- Match each expression to either \((A-B)^2\) or \((A+B)^2\). - Square each term and include twice their product. - Apply the power-of-a-power rule when squaring \(x^2\).

1. For \(\left(\frac{2}{3}x-6\right)^2\), the square terms are \(\frac{4}{9}x^2\) and \(36\), and the middle term is \(-2\left(\frac{2}{3}x\right)(6)=-8x\). Thus the result is \(\frac{4}{9}x^2-8x+36\). 2. For \((x^2+4y)^2\), the square terms are \(x^4\) and \(16y^2\), and the middle term is \(2(x^2)(4y)=8x^2y\). Thus the result is \(x^4+8x^2y+16y^2\).

1) \(\frac{4}{9}x^2-8x+36\) 2) \(x^4+8x^2y+16y^2\)

- Expand each product before combining terms. - Recognize both a binomial square and a product of a sum and difference. - In part b), distribute the negative factor carefully.

1. a) Expand the binomial square and the product of a sum and difference: \(4(y^2-2y+1)+(y^2-4)\). 2. Distribute and combine like terms: \(4y^2-8y+4+y^2-4=5y^2-8y\). 3. b) Expand both parts: \((3a+2)^2=9a^2+12a+4\) and \(-3(a^2-4)=-3a^2+12\). 4. Combine like terms: \(9a^2+12a+4-3a^2+12=6a^2+12a+16\).

a) \(5y^2-8y\) b) \(6a^2+12a+16\)

- Expand the two binomial squares first. - Distribute the subtraction sign across the second polynomial. - Simplify the expression in parentheses before multiplying by the outside factor.

1. Expand the binomial squares: \((2r+3s)^2=4r^2+12rs+9s^2\) and \((2r-3s)^2=4r^2-12rs+9s^2\). 2. Subtract: \((4r^2+12rs+9s^2)-(4r^2-12rs+9s^2)=24rs\). 3. Multiply by the outside factor: \((24rs)\left(\frac{1}{2}rs\right)=12r^2s^2\).

\(12r^2s^2\)

- Group repeated sums or differences into a single expression. - Look for the structure \((A+B)(A-B)\). - When subtracting a binomial square, distribute the negative sign to every term.

1. a) Group the terms as \(((r+s)+(t+u))((r+s)-(t+u))\). Then \((r+s)^2-(t+u)^2\). 2. Expand both squares: \(r^2+2rs+s^2-(t^2+2tu+u^2)=r^2+2rs+s^2-t^2-2tu-u^2\). 3. b) Rewrite the factors as \((10-(x-2y))(10+(x-2y))\). Then the product is \(100-(x-2y)^2\). 4. Expand and simplify: \(100-(x^2-4xy+4y^2)=100-x^2+4xy-4y^2\).

a) \(((r+s)+(t+u))((r+s)-(t+u))=(r+s)^2-(t+u)^2\) \(=r^2+2rs+s^2-t^2-2tu-u^2\) b) \((10-(x-2y))(10+(x-2y))=100-(x-2y)^2\) \(=100-x^2+4xy-4y^2\)

- Expand both binomial squares first. - Apply the outside factor to every term. - Distribute the subtraction sign across the second polynomial.

1. Expand the binomial squares: \((a+5)^2=a^2+10a+25\) and \((a-4)^2=a^2-8a+16\). 2. Substitute and distribute: \(2(a^2+10a+25)-(a^2-8a+16)\). 3. Remove parentheses: \(2a^2+20a+50-a^2+8a-16\). 4. Combine like terms to get \(a^2+28a+34\).

\(a^2+28a+34\)

- Expand each product separately. - Recognize the product \((x+y)(x-y)\). - Distribute \(-2x\) to both terms in the final factor. - Combine like terms only after all products are expanded.

1. Expand the first square: \((3x-2y)^2=9x^2-12xy+4y^2\). 2. Use the difference-of-squares identity: \(3(x+y)(x-y)=3x^2-3y^2\). 3. Distribute in the last term: \(-2x(x-4y)=-2x^2+8xy\). 4. Combine like terms: \((9+3-2)x^2+(-12+8)xy+(4-3)y^2=10x^2-4xy+y^2\).

\(10x^2-4xy+y^2\)

- Look for products in the form \((A-B)(A+B)\). - In part a), simplify two factors at a time. - In part b), expand the square before combining like terms.

1. a) First, \((x-3)(x+3)=x^2-9\). 2. Then \((x^2-9)(x^2+9)=x^4-81\). 3. b) Expand the square: \((k+4)^2=k^2+8k+16\). 4. Subtract \(8k\): \(k^2+8k+16-8k=k^2+16\).

a) \((x-3)(x+3)=x^2-9\), then \((x^2-9)(x^2+9)=x^4-81\) b) \((k+4)^2=k^2+8k+16\), so \((k+4)^2-8k=k^2+16\)

- Work from left to right, combining two factors at a time. - Look for the same sum-and-difference pattern at every stage. - Apply the power-of-a-power rule when squaring a power.

1. Multiply the first two factors: \((y-2)(y+2)=y^2-4\). 2. Multiply by the third factor: \((y^2-4)(y^2+4)=y^4-16\). 3. Multiply by the fourth factor: \((y^4-16)(y^4+16)=y^8-256\).

\((y-2)(y+2)=y^2-4\) \((y^2-4)(y^2+4)=y^4-16\) \((y^4-16)(y^4+16)=y^8-256\)

- Group the nonmatching terms into one expression. - Rewrite each product in the form \((A+B)(A-B)\). - Distribute the negative sign after expanding the grouped square.

1. a) Group the factors as \((m+(2n+3p))(m-(2n+3p))\). This equals \(m^2-(2n+3p)^2\). 2. Expand and simplify: \(m^2-(4n^2+12np+9p^2)=m^2-4n^2-12np-9p^2\). 3. b) Group the factors as \((4u+(v-5w))(4u-(v-5w))\). This equals \(16u^2-(v-5w)^2\). 4. Expand and simplify: \(16u^2-(v^2-10vw+25w^2)=16u^2-v^2+10vw-25w^2\).

a) \((m+(2n+3p))(m-(2n+3p))=m^2-(2n+3p)^2\) \(=m^2-4n^2-12np-9p^2\) b) \((4u+(v-5w))(4u-(v-5w))=16u^2-(v-5w)^2\) \(=16u^2-v^2+10vw-25w^2\)

- Identify the sum-and-difference product. - Expand the binomial square separately. - Distribute the subtraction sign before combining like terms.

1. Apply the difference-of-squares identity: \((3x+2y)(3x-2y)=9x^2-4y^2\). 2. Expand the binomial square: \((3x-y)^2=9x^2-6xy+y^2\). 3. Subtract and combine like terms: \(9x^2-4y^2-(9x^2-6xy+y^2)=6xy-5y^2\).

\(6xy-5y^2\)

- Expand both squares inside the parentheses first. - Distribute the subtraction sign carefully. - Apply the product rule for exponents in the final multiplication.

1. Expand the squares: \((b^2+4)^2=b^4+8b^2+16\) and \((b^2-4)^2=b^4-8b^2+16\). 2. Subtract: \((b^4+8b^2+16)-(b^4-8b^2+16)=16b^2\). 3. Multiply by the outside factor: \(\frac{1}{4}b(16b^2)=4b^3\).

\(4b^3\)

- Match each expression to a sum-square or difference-square identity. - Square coefficients and variable powers separately. - For each middle term, multiply twice the product of the two binomial terms.

1. \((0.5x^2+0.2y^3)^2=0.25x^4+0.2x^2y^3+0.04y^6\). 2. \(\left(\frac{3}{4}a-\frac{1}{3}b\right)^2=\frac{9}{16}a^2-\frac{1}{2}ab+\frac{1}{9}b^2\). 3. \((1.2p^2-0.5q)^2=1.44p^4-1.2p^2q+0.25q^2\).

1) \(0.25x^4+0.2x^2y^3+0.04y^6\) 2) \(\frac{9}{16}a^2-\frac{1}{2}ab+\frac{1}{9}b^2\) 3) \(1.44p^4-1.2p^2q+0.25q^2\)

- Expand the two products separately. - Keep the second product in parentheses when subtracting it. - Combine the \(a^2\)-terms, \(a\)-terms, and constants separately. - Substitute only after simplifying completely.

1. Expand the first product: \(2a(3a - 1) = 6a^2 - 2a\). 2. Expand the second product: \((a + 2)(6a - 5) = 6a^2 + 7a - 10\). 3. Subtract and combine like terms: \(6a^2 - 2a - (6a^2 + 7a - 10) = -9a + 10\). 4. Evaluate: \(-9 \cdot \left(-\frac{2}{3}\right) + 10 = 16\).

\(16\)

- Multiply every term in the first binomial by every term in the second. - Write each decimal as \(1\) plus a small amount. - The error is the difference between the exact product and the estimate.

1. Expand: \((1 + a)(1 + b) = 1 + a + b + ab\). The approximation omits \(ab\). 2. Here \(a = 0.002\) and \(b = 0.006\), so the estimate is \(1 + 0.002 + 0.006 = 1.008\). 3. The exact product is \(1.002 \cdot 1.006 = 1.008012\). 4. The difference is \(1.008012 - 1.008 = 0.000012\).

a) \((1 + a)(1 + b) = 1 + a + b + ab\); the omitted term is \(ab\) b) \(1.008\) c) Exact product: \(1.008012\); difference: \(0.000012\)

- Distribute each factor across its parentheses. - Watch the signs as you combine like terms. - Look for terms that cancel. - Substitute the values after simplifying.

1. Distribute: \(4a(a - 2b) = 4a^2 - 8ab\) and \(2b(4a + b) = 8ab + 2b^2\). 2. Combine the expressions: \(4a^2 - 8ab + 8ab + 2b^2 - 3a^2\). 3. The opposite \(ab\)-terms cancel, so \(T(a,b) = a^2 + 2b^2\). 4. Substitute: \((1.2)^2 + 2 \cdot (-0.5)^2 = 1.44 + 2 \cdot 0.25 = 1.94\).

Simplified expression: \(a^2 + 2b^2\) Value: \(1.94\)

- Simplify each expression separately and compare the results. - Remember that \(xy = yx\). - A minus sign before parentheses changes the sign of each term when the parentheses are removed. - Substitute into the simpler equivalent expression.

1. Simplify \(A\): \(A = 2x^2 + xy - xy + 3y^2 = 2x^2 + 3y^2\). 2. Simplify \(B\): \(B = 2x^2 + 2y^2 + y^2 = 2x^2 + 3y^2\). 3. Because both expressions simplify to \(2x^2 + 3y^2\), they are equivalent. 4. Substitute: \(2 \cdot (-5)^2 + 3 \cdot \left(\frac{1}{3}\right)^2 = 50 + \frac{1}{3} = \frac{151}{3}\).

a) Both expressions simplify to \(2x^2 + 3y^2\). b) \(\frac{151}{3}\), or \(50\frac{1}{3}\)

- Combine only terms with identical variable parts. - Divide each remaining term by the monomial. - Determine the sign, coefficient, and variable power separately.

1. Combine like terms: \(0.5a^2 - 1.1a^2 = -0.6a^2\). 2. The dividend becomes \(-0.2a^3 - 0.6a^2 + 0.04a\). 3. Divide each term by \(-0.2a\): \(a^2 + 3a - 0.2\).

\(a^2 + 3a - 0.2\)

- Use the order of operations. - Divide each term inside the parentheses by \(3\). - A minus sign before parentheses changes both signs when the parentheses are removed. - Combine only terms with the same variable part.

1. Distribute: \(3x(x - 4) = 3x^2 - 12x\). 2. Divide each term: \((6x^2 + 9x) \div 3 = 2x^2 + 3x\). 3. Subtract and combine like terms: \(3x^2 - 12x - (2x^2 + 3x) = x^2 - 15x\).

\(x^2 - 15x\)

- Cutting a square from both ends reduces each base dimension by twice the cutout side. - Expand the square before subtracting areas. - Use the solved side length to determine the base dimensions and height.

1. The original area is \(a^2\). 2. The base side length is \(a-8\), so the base area is \((a-8)^2=a^2-16a+64\). 3. The area difference is \(a^2-(a^2-16a+64)=16a-64\). 4. Set \(16a-64=160\), so \(a=14\). 5. The base is \(6\,\text{in.}\) by \(6\,\text{in.}\), and the height is \(4\,\text{in.}\). 6. The volume is \(6\cdot6\cdot4=144\,\text{in.}^3\).

a) Original area: \(a^2\) Base area: \((a-8)^2=a^2-16a+64\) Area difference: \(16a-64\) b) \(a=14\,\text{in.}\); volume \(=144\,\text{in.}^3\)

- Express both original dimensions in terms of \(b\). - Apply both dimension changes before writing the redesigned area. - Expand both products completely. - Subtract the original polynomial from the redesigned polynomial.

1. The original length is \(b+8\), so the original area is \(A_1=b(b+8)=b^2+8b\). 2. The redesigned dimensions are \(b+3\) and \(b+5\), so \(A_2=(b+3)(b+5)=b^2+8b+15\). 3. Subtract: \(A_2-A_1=(b^2+8b+15)-(b^2+8b)=15\). 4. Therefore, the area increases by \(15\,\text{m}^2\).

The area increases by \(15\,\text{m}^2\).

- The mean of two values is their sum divided by \(2\). - Express the square's side length in terms of \(b\). - Expand both area expressions before comparing. - Identify the term that differs between the two results.

1. The rectangular lot has width \(b\) and length \(b+12\), so \(A_R=b(b+12)=b^2+12b\). 2. The square's side length is the mean: \(s=\frac{b+(b+12)}{2}=b+6\). 3. Its area is \(A_S=(b+6)^2=b^2+12b+36\). 4. Since \(A_S-A_R=36\), the square lot has \(36\,\text{m}^2\) more area.

a) Rectangle: \(b(b+12)=b^2+12b\); square: \((b+6)^2=b^2+12b+36\) b) The square parking lot has \(36\,\text{m}^2\) more area.

- Terms are like terms only when all variable bases and exponents match. - A term without a written coefficient has coefficient \(1\). - Track subtraction signs carefully.

1. \(4x^2 + x^2 - 2x^2 = 3x^2\). 2. \(a^2b + a^2b + ab^2 = 2a^2b + ab^2\). 3. \(3z^3 + z^3 - 2z^3 = 2z^3\). 4. \(2p^2 + q^3 + p^2 + q^3 + p^2 = 4p^2 + 2q^3\).

1) \(3x^2\) 2) \(2a^2b + ab^2\) 3) \(2z^3\) 4) \(4p^2 + 2q^3\)

- Check how each sign changes when the second polynomial is subtracted. - Subtracting a negative term produces a positive term. - Determine the sign of the \(9x\) term after removing the parentheses.

1. Lucas did not reverse the sign of \(-9x\) when subtracting the second polynomial. He treated \(-4x-9x\) as the linear part instead of \(-4x+9x\). 2. Subtract the second polynomial correctly: \((15x^2-4x)-(7x^2-9x)=15x^2-4x-7x^2+9x\). 3. Combine like terms: \(15x^2-7x^2=8x^2\) and \(-4x+9x=5x\). 4. The correct result is \(8x^2+5x\).

Lucas failed to change the sign of \(-9x\) when he subtracted the second polynomial. The correct result is \(8x^2+5x\).

- Multiply the outside factor by every term inside the parentheses. - Add exponents when multiplying powers with the same base. - Track the negative sign during distribution.

1. Distribute and add exponents: \(4a^{n+3} - 8a^{2n}\). 2. Distribute: \(-6x^{k+(k+1)} - 12x^{k+1} = -6x^{2k+1} - 12x^{k+1}\). 3. Distribute: \(3y^{m+2m} - 2y^{m+1} = 3y^{3m} - 2y^{m+1}\).

1) \(4a^{n+3} - 8a^{2n}\) 2) \(-6x^{2k+1} - 12x^{k+1}\) 3) \(3y^{3m} - 2y^{m+1}\)

- Write the two new side lengths in terms of \(s\). - Expand the rectangle area before comparing it with the square. - The \(s^2\) terms cancel in the difference.

1. The square area is \(s^2\). 2. The rectangle area is \((s+2)(s+5)=s^2+7s+10\). 3. The area difference is \(7s+10\). 4. Set \(7s+10=45\), so \(s=5\).

a) Square area: \(s^2\) Rectangle area: \(s^2+7s+10\) b) Area difference: \(7s+10\) \(7s+10=45\), so \(s=5\,\text{in.}\).

- Express the length in terms of \(b\). - Expand both area products before subtracting. - Simplify the area-increase polynomial. - Use the given increase only after the polynomial difference has been formed.

1. The original length is \(b+5\), so the original area is \(b(b+5)=b^2+5b\). 2. The new area is \((b+2)(b+5)=b^2+7b+10\). 3. The increase is \((b^2+7b+10)-(b^2+5b)=2b+10\). 4. Set \(2b+10=30\). Then \(2b=20\), so \(b=10\). 5. The original length is \(10+5=15\,\text{ft}\).

The garden was originally \(10\,\text{ft}\) wide and \(15\,\text{ft}\) long.

- The middle term is twice the product of the binomial terms. - Use square roots of the known square terms. - Compare each identity with \(A^2\pm2AB+B^2\).

1. a) Since \(30a^2b^3=2\cdot3a^2\cdot5b^3\), the missing binomial term is \(5b^3\). Its square is \(25b^6\). 2. b) Since \(4x^6=(2x^3)^2\), the missing first term is \(2x^3\). The final square term is \(\left(\frac{1}{2}y^2\right)^2=\frac{1}{4}y^4\).

a) \((3a^2+5b^3)^2=9a^4+30a^2b^3+25b^6\) b) \(\left(2x^3-\frac{1}{2}y^2\right)^2=4x^6-2x^3y^2+\frac{1}{4}y^4\)

- Use multiplication as the inverse of division. - Multiply every term in the quotient by the divisor. - Use the given middle term to check the pattern.

1. Multiply each term on the right by \(4xy\). 2. The first missing term is \(2x^2(4xy)=8x^3y\). 3. The middle term checks: \(-3xy(4xy)=-12x^2y^2\). 4. The final missing term is \(5y^2(4xy)=20xy^3\). 5. Thus, \((8x^3y-12x^2y^2+20xy^3)\div4xy=2x^2-3xy+5y^2\).

The missing terms are \(8x^3y\) and \(20xy^3\).

- Multiply each term in the first factor by every term in the second factor. - Track the signs carefully. - Look for opposite terms that cancel.

1. Distribute: \((x + 3)(x^2 - 3x + 9) = x(x^2 - 3x + 9) + 3(x^2 - 3x + 9)\). 2. Multiply: \(x^3 - 3x^2 + 9x + 3x^2 - 9x + 27\). 3. The quadratic terms and linear terms cancel, leaving \(x^3 + 27\). Therefore, the identity is true.

The left side simplifies to \(x^3 + 27\), so the identity is verified.

- Check the exponents of \(x\). - Expand products before identifying degree and leading coefficient. - Distinguish a power function from an exponential function. - Real coefficients may include fractions and irrational numbers.

1. A polynomial has real coefficients and only nonnegative integer powers of \(x\). 2. For a), the highest power is \(3\), so it is a degree-\(3\) polynomial with leading coefficient \(0.2\). 3. For b), the variable appears in a denominator, so the function is not polynomial. 4. For c), expand: \((x^2-1)(2-x)=-x^3+2x^2+x-2\). It is degree \(3\) with leading coefficient \(-1\). 5. For d), \(3^x\) is exponential, so the function is not polynomial. 6. For e), \(g_5(x)=\frac{1}{8}x^4-\pi x\). It is degree \(4\) with leading coefficient \(\frac{1}{8}\).

a) Polynomial; degree \(3\); leading coefficient \(0.2\) b) Not polynomial c) Polynomial; degree \(3\); leading coefficient \(-1\) d) Not polynomial e) Polynomial; degree \(4\); leading coefficient \(\frac{1}{8}\)

- Focus first on the leading term of each polynomial rather than expanding every term. - Ask whether any other product term could have the same exponent as the product of the two leading terms. - For the sum, combine the highest-degree terms before deciding the degree.

1. The leading terms of \(P\) and \(Q\) are \(3x^4\) and \(-2x^3\). Their product is \(-6x^7\), which cannot be canceled by any other product term. Therefore, \(P(x)Q(x)\) has degree \(7\). 2. \(R(x)+S(x)=(5x^4-5x^4)+3x^3+2x^2+3=3x^3+2x^2+3\), so the sum has degree \(3\). 3. For a product of nonzero polynomials, the leading terms multiply to a nonzero leading term, so the degrees add. In a sum, leading terms of the same degree can cancel, which can lower the degree.

a) Degree \(7\) b) Degree \(3\) c) Degrees add in a product of nonzero polynomials, but leading-term cancellation can lower the degree of a sum.

- Identify the square roots of the first and last terms. - The middle term is twice the product of the binomial terms. - Use the sign in the binomial to determine the middle sign.

1. a) Since \(25 = 5^2\), the missing binomial term is \(5\). The middle term is \(2(3x)(5) = 30x\). 2. b) Since \(0.16x^2 = (0.4x)^2\), the missing first term is \(0.4x\). The middle term is \(2(0.4x)(0.2y^2) = 0.16xy^2\).

a) \((3x + 5)^2 = 9x^2 + 30x + 25\) b) \((0.4x - 0.2y^2)^2 = 0.16x^2 - 0.16xy^2 + 0.04y^4\)

- Find the square roots of the first and last terms. - Check that twice their product matches the middle term. - Compare the sum-square and difference-square patterns.

1. The first term is \(\left(\frac{1}{2}a^3\right)^2\), and the last term is \((3b^2)^2\). 2. Their doubled product is \(2\left(\frac{1}{2}a^3\right)(3b^2) = 3a^3b^2\), so the original binomial is \(\left(\frac{1}{2}a^3 + 3b^2\right)^2\). 3. To make the middle term negative, replace the plus sign with a minus sign: \(\left(\frac{1}{2}a^3 - 3b^2\right)^2\).

a) \(\left(\frac{1}{2}a^3 + 3b^2\right)^2\) b) \(\left(\frac{1}{2}a^3 - 3b^2\right)^2\); changing the sign changes only the middle term’s sign.

- In Case II, the second number is less than \(1\), so \(b\) is negative. - Compare the exact expansion with the approximation to identify the error term. - Compute the absolute difference between each estimate and exact product.

1. Case I has \(a = 0.003\), \(b = 0.003\), so the estimate is \(1.006\). 2. Case II has \(a = 0.003\), \(b = -0.003\), so the estimate is \(1.000\). 3. The exact products are \(1.003 \cdot 1.003 = 1.006009\) and \(1.003 \cdot 0.997 = 0.999991\). 4. Each absolute error is \(0.000009\). The omitted products are \(0.003 \cdot 0.003 = 0.000009\) and \(0.003 \cdot (-0.003) = -0.000009\), which have the same absolute value.

a) Case I: \(1.006\); Case II: \(1.000\) b) Case I: \(1.006009\); Case II: \(0.999991\) c) Both absolute errors are \(0.000009\) because the omitted terms \(ab\) have equal absolute values.

- Express the original length as twice the width. - Each base dimension loses \(4\,\text{in.}\) in total. - Expand the base-area product and subtract it from the original area.

1. The original sheet is \(b\) by \(2b\), so its area is \(2b^2\). 2. The base dimensions are \(b-4\) and \(2b-4\), so the base area is \((b-4)(2b-4)=2b^2-12b+16\). 3. The area difference is \(2b^2-(2b^2-12b+16)=12b-16\). 4. Set \(12b-16=44\), so \(b=5\). 5. The original sheet is \(5\,\text{in.}\) by \(10\,\text{in.}\). 6. The base is \(1\,\text{in.}\) by \(6\,\text{in.}\), with height \(2\,\text{in.}\), so the volume is \(12\,\text{in.}^3\).

a) Original area: \(2b^2\) Base area: \(2b^2-12b+16\) Area difference: \(12b-16\) b) Original dimensions: \(5\,\text{in.}\) by \(10\,\text{in.}\) Volume: \(12\,\text{in.}^3\)

- Group the terms into two sum-and-difference pairs. - Use \((u+v)^2+(u-v)^2=2u^2+2v^2\). - Simplify the remaining difference of two binomial squares.

1. Group the first pair using \(u=a+b\): \((u+c)^2+(u-c)^2=2u^2+2c^2=2(a+b)^2+2c^2\). 2. Group the second pair using \(v=a-b\): \((v+c)^2+(v-c)^2=2v^2+2c^2=2(a-b)^2+2c^2\). 3. Subtract the pairs: \(2(a+b)^2-2(a-b)^2=2\left((a+b)^2-(a-b)^2\right)\). 4. Since \((a+b)^2-(a-b)^2=4ab\), the expression equals \(2\cdot4ab=8ab\).

Let \(u=a+b\) and \(v=a-b\). \((u+c)^2+(u-c)^2=2u^2+2c^2\) \((v+c)^2+(v-c)^2=2v^2+2c^2\) Therefore, the left side is \(2(a+b)^2-2(a-b)^2\) \(=2\big((a+b)^2-(a-b)^2\big)\) \(=2(4ab)=8ab\). Thus the identity holds for all real \(a,b,c\).