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) \(0\)

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

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

- 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) \(0\)

- 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) \(\frac{1}{2}b^2\)

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

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

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

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

- 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) \(y^4 - y^2 - 12\)

- 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) \(4k+3\)

- 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: \(f_5(x)=\sqrt{7}x^3\)

- Represent the square's side length with a variable. - Express both rectangle dimensions using that variable. - Set the two area expressions equal. - Expand the product and check whether the quadratic terms cancel.

1. Let \(s\) feet be the side length of the square. Its area is \(s^2\). 2. The rectangle has dimensions \(s + 4\) and \(s - 3\), so its area is \((s + 4)(s - 3)\). 3. Set the areas equal: \(s^2 = (s + 4)(s - 3)\). 4. Expand the product: \(s^2 = s^2 + s - 12\). 5. Subtract \(s^2\) from both sides: \(0 = s - 12\), so \(s = 12\).

The square bed has side length \(12\,\text{ft}\).

- 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) \(34 - 7x\)

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

- 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: \(x^2 - 10x + 25 - x^2 - 4x - 4 = -14x + 21\).

\(-14x + 21\)

- 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) \(18y^2 - 12y\)

- 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) \(k^2 + m\)

- 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) \(5a^2 + 6a\)

- 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) \(-12x + 30\)

- 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) \(4s^4t^3 - 12s^3t^4 + 8s^2t^5\)

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

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

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

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

- 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) \(-0.5\)

- 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.

- 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 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 \(3\frac{1}{3}\)

- 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) \(-2x^4 + x^3 - 5x^2\)

- 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 length in terms of the width. - Write the area before and after the change. - Set the difference between the areas equal to \(12\). - Visualize the rectangular strip that is removed.

1. Let \(w\) inches be the width. The length is \(3w\) inches, so the original area is \(3w^2\). 2. After the length is shortened, the new area is \((3w - 2)w = 3w^2 - 2w\). 3. The area decreases by \(12\,\text{in.}^2\), so \(3w^2 - (3w^2 - 2w) = 12\). 4. Simplify: \(2w = 12\), so \(w = 6\). 5. The original length is \(3 \cdot 6\,\text{in.} = 18\,\text{in.}\). 6. The removed region is a strip \(2\,\text{in.}\) long and \(w\) inches wide, so its area is \(2w\).

a) The original width is \(6\,\text{in.}\), and the original length is \(18\,\text{in.}\). b) Shortening only the length removes a \(2\,\text{in.}\)-by-\(w\) strip, so the area decrease is \(2w\).

- Write an area expression for each extended bed. - Subtract the smaller new area from the larger new area. - Look for terms that cancel. - Set the remaining 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. The difference in area is \((a^2 + 5a) - (a^2 + 2a) = 3a\). 4. Set the difference equal to \(15\): \(3a = 15\). 5. Divide by \(3\): \(a = 5\).

The original side length of each square garden bed 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 \(\frac{1}{2}a^2 - \frac{11}{72}ab - \frac{4}{3}a - \frac{1}{6}b^2 + b\).

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

- 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 \(x = 0.75\): \(-4\)

- 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 dimensions using one variable. - Determine the common side length after the changes. - Write and compare the two area expressions. - Expand the square and simplify the equation.

1. Let \(x\) inches be the original width. The original length is \(x + 7\), so the original 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 square's area is \(55\,\text{in.}^2\) greater: \(x^2 + 10x + 25 = x^2 + 7x + 55\). 4. Subtract \(x^2 + 7x\): \(3x + 25 = 55\). 5. Subtract \(25\): \(3x = 30\), so \(x = 10\). 6. The square's side length is \(x + 5 = 15\,\text{in.}\).

The square''s side length is \(15\,\text{in.}\).

- Express the original length in terms of the width. - Write the new dimensions after both changes. - Set the original and new area expressions equal. - Use the original dimensions to calculate the requested perimeter.

1. Let \(b\) inches be the original width. The original length is \(b + 8\), 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. Set the areas equal: \(b^2 + 8b = b^2 + 7b + 10\). 4. Subtract \(b^2 + 7b\): \(b = 10\). 5. The original dimensions are \(10\,\text{in.}\) and \(18\,\text{in.}\). 6. The perimeter is \(2(10 + 18) = 56\,\text{in.}\).

The original rectangle''s perimeter is \(56\,\text{in.}\).

- Express the original length in terms of the width. - Write area expressions before and after the changes. - Relate the two areas using the stated increase. - Expand the product and look for quadratic terms that cancel.

1. Let \(w\) inches be the original width. The original length is \(3w\), so the original area is \(3w^2\). 2. The new dimensions are \(w + 2\) and \(3w - 3\), so the new area is \((w + 2)(3w - 3)\). 3. The area increases by \(15\,\text{in.}^2\), so \((w + 2)(3w - 3) = 3w^2 + 15\). 4. Expand: \(3w^2 + 3w - 6 = 3w^2 + 15\). 5. Subtract \(3w^2\) and add \(6\): \(3w = 21\), so \(w = 7\). 6. The original length is \(3 \cdot 7\,\text{in.} = 21\,\text{in.}\).

The original rectangle was \(7\,\text{in.}\) wide and \(21\,\text{in.}\) long.

- 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 in terms of the width. - Write the original and new area expressions. - Set the new area equal to the original area plus \(41\). - Expand carefully and simplify.

1. Let \(x\) feet be the original width. The original length is \(x + 5\), so the original area is \(x(x + 5) = x^2 + 5x\). 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 area increases by \(41\,\text{ft}^2\), so \(x^2 + 10x + 16 = x^2 + 5x + 41\). 4. Subtract \(x^2 + 5x\): \(5x + 16 = 41\). 5. Subtract \(16\): \(5x = 25\), so \(x = 5\). 6. The original length is \(5\,\text{ft} + 5\,\text{ft} = 10\,\text{ft}\).

The garden bed was originally \(5\,\text{ft}\) wide and \(10\,\text{ft}\) long.

- 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 \(4x^2y\).

\(4x^2y\)

- 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: \((24rs)\left(\frac{1}{2}rs\right)=12r^2s^2\).

\(12r^2s^2\)

- 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: \((9+3-2)x^2+(-12+8)xy+(4-3)y^2=10x^2-4xy+y^2\).

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

- 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: \(\frac{1}{4}b(16b^2)=4b^3\).

\(4b^3\)

- 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: \(3x^2 - 12x - (2x^2 + 3x) = x^2 - 15x\).

\(x^2 - 15x\)

- Identify how the corner cutouts change each base dimension. - Subtract twice the cutout length from each original dimension. - Compare the area of the original sheet with the area of the box's base. - Use base area times height for the box's volume.

1. Let \(a\) inches be the side length of the original square sheet. Its area is \(a^2\). 2. Cutting \(4\,\text{in.}\) from both ends of each dimension leaves a square base with side length \(a - 8\). Its area is \((a - 8)^2\). 3. The area difference is \(160\,\text{in.}^2\), so \(a^2 - (a - 8)^2 = 160\). 4. Expand: \(a^2 - (a^2 - 16a + 64) = 160\). 5. Simplify: \(16a - 64 = 160\), so \(16a = 224\) and \(a = 14\). 6. The base is \(6\,\text{in.}\) by \(6\,\text{in.}\), and the height is \(4\,\text{in.}\). Thus, the volume is \(6 \cdot 6 \cdot 4 = 144\,\text{in.}^3\).

The original sheet has side length \(14\,\text{in.}\), and the box has volume \(144\,\text{in.}^3\).

- Let a variable represent the original width. - Write the original and new dimensions in terms of that variable. - Expand both area expressions. - Subtract the original area from the new area.

1. Let the original width be \(b\). Then the original length is \(b+8\), and the original area is \(A_1=b(b+8)=b^2+8b\). 2. The new width is \(b+3\), and the new length is \((b+8)-3=b+5\). 3. The new area is \(A_2=(b+3)(b+5)=b^2+8b+15\). 4. The change is \(A_2-A_1=(b^2+8b+15)-(b^2+8b)=15\). 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 \(8x^2+5x\).

- Write the area of the square. - Express both rectangle dimensions in terms of \(s\). - Set the new area equal to the old area plus \(45\). - Expand the product and simplify.

1. The square's area is \(s^2\). 2. The rectangle has dimensions \(s + 2\) and \(s + 5\), so its area is \((s + 2)(s + 5)\). 3. Write the area equation: \((s + 2)(s + 5) = s^2 + 45\). 4. Expand: \(s^2 + 7s + 10 = s^2 + 45\). 5. Subtract \(s^2\): \(7s + 10 = 45\). 6. Subtract \(10\) and divide by \(7\): \(7s = 35\), so \(s = 5\).

The square''s side length is \(5\,\text{in.}\).

- Express the length in terms of the width. - Write area expressions before and after the width changes. - Subtract the original area from the new area. - Set the difference equal to \(30\).

1. Let \(b\) feet be the original width. The length is \(b + 5\) feet. 2. The original area is \(b(b + 5)\). 3. The new area is \((b + 2)(b + 5)\). 4. The increase is \(30\,\text{ft}^2\), so \((b + 2)(b + 5) - b(b + 5) = 30\). 5. Expand and simplify: \(b^2 + 7b + 10 - b^2 - 5b = 30\), so \(2b + 10 = 30\). 6. Subtract \(10\) and divide by \(2\): \(b = 10\). 7. The original length is \(10\,\text{ft} + 5\,\text{ft} = 15\,\text{ft}\).

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

- 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, \((8x^3y - 12x^2y^2 + 20xy^3) \div 4xy = 2x^2 - 3xy + 5y^2\).

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 \(\frac{1}{8}\)

- Express the original length in terms of the height. - Write the original and new area expressions. - Set the new area equal to the original area plus \(30\). - Expand the product and simplify; the quadratic terms will cancel.

1. Let \(x\) inches be the original height. 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 area increases by \(30\), so \(2x^2 + 4x - 6 = 2x^2 + 30\). 5. Subtract \(2x^2\): \(4x - 6 = 30\). Add \(6\): \(4x = 36\). Divide by \(4\): \(x = 9\). 6. The original height was \(9\,\text{in.}\), and the original length was \(18\,\text{in.}\).

The original dimensions were \(9\,\text{in.}\) by \(18\,\text{in.}\).

- Represent the shorter side with a variable and express the longer side in terms of it. - Write expressions for the original and new areas. - Set the new area equal to the original area plus \(66\). - Expand both products and check whether the quadratic terms cancel.

1. Let \(x\) inches be the shorter side. The longer side is \(x + 5\) inches. 2. The original area is \(x(x + 5)\). 3. The new side lengths are \(x + 3\) and \(x + 7\), so the new area is \((x + 3)(x + 7)\). 4. The area increases by \(66\,\text{in.}^2\), so \((x + 3)(x + 7) = x(x + 5) + 66\). 5. Expand: \(x^2 + 10x + 21 = x^2 + 5x + 66\). 6. Subtract \(x^2\) and solve: \(10x + 21 = 5x + 66\), so \(5x = 45\) and \(x = 9\). 7. The longer side is \(9\,\text{in.} + 5\,\text{in.} = 14\,\text{in.}\).

The original side lengths are \(9\,\text{in.}\) and \(14\,\text{in.}\).

- 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) \(x\)

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

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

- 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 \(n = -1\frac{1}{8}\): \(-17\)

- Consecutive odd integers differ by \(2\). - Write one product for each pair of odd integers. - Expand both products and distribute the subtraction carefully. - Observe what happens to the \(x^2\) terms.

1. Let \(x\) be the first odd integer. The next three are \(x + 2\), \(x + 4\), and \(x + 6\). 2. Write the product-difference equation \((x + 4)(x + 6) - x(x + 2) = 48\). 3. Expand both products: \(x^2 + 10x + 24 - (x^2 + 2x) = 48\). 4. Distribute the subtraction and combine like terms: \(x^2 + 10x + 24 - x^2 - 2x = 48\), so \(8x + 24 = 48\). 5. Subtract \(24\): \(8x = 24\). Divide by \(8\): \(x = 3\). 6. The four integers are \(3\), \(5\), \(7\), and \(9\).

The four odd integers are \(3\), \(5\), \(7\), and \(9\).

- Expand each product using the distributive property. - A minus sign before parentheses changes every sign inside. - After simplifying the equation, isolate the variable term. - An expression independent of a variable simplifies to a constant.

1. Expand and subtract: \((n^2 + 7n + 12) - (n^2 + 7n + 6) = 6\). Since \(n\) cancels, the value is independent of \(n\). 2. Expand both products: \((2x + 1)(3x - 2) = 6x^2 - x - 2\) and \((x - 1)(x + 2) = x^2 + x - 2\). 3. Substitute and simplify: \(6x^2 - x - 2 - 6(x^2 + x - 2) = 31\), so \(-7x + 10 = 31\). 4. Solve: \(-7x = 21\), so \(x = -3\).

1) \(6\) 2) \(x = -3\)

- Write area expressions for the square and the redesigned rectangle. - The quadratic terms should cancel when you compare the areas. - For part b), calculate both areas using the value from part a). - A single counterexample is enough to disprove an “always” claim.

1. Let \(s\) feet be the square's side length. Its area is \(s^2\). 2. The redesigned rectangle has dimensions \(s + 6\) and \(s - 4\), so its area is \((s + 6)(s - 4) = s^2 + 2s - 24\). 3. The new area is \(16\,\text{ft}^2\) greater, so \(s^2 + 2s - 24 = s^2 + 16\). 4. Subtract \(s^2\) and solve: \(2s - 24 = 16\), so \(2s = 40\) and \(s = 20\). 5. For part b), the original area is \(20 \cdot 20 = 400\,\text{ft}^2\). 6. Changing the sides by \(5\,\text{ft}\) gives dimensions \(25\,\text{ft}\) and \(15\,\text{ft}\), with area \(25 \cdot 15 = 375\,\text{ft}^2\). 7. Since \(375 \neq 400\), the claim is false.

a) The original side length was \(20\,\text{ft}\). b) The changed dimensions are \(25\,\text{ft}\) by \(15\,\text{ft}\), with area \(375\,\text{ft}^2\), not \(400\,\text{ft}^2\). Therefore, the claim is false.

- 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 length in terms of the width. - Each base dimension is reduced by twice the cutout length. - Compare the original area with the base area. - Check that the resulting base dimensions are positive before finding the volume.

1. Let \(b\) inches be the width. The length is \(2b\), so the original area is \(2b^2\). 2. The box's base has dimensions \(2b - 4\) and \(b - 4\), so its area is \((2b - 4)(b - 4)\). 3. Expand the base area: \((2b - 4)(b - 4) = 2b^2 - 12b + 16\). 4. The area difference is \(44\,\text{in.}^2\), so \(2b^2 - (2b^2 - 12b + 16) = 44\). 5. Simplify: \(12b - 16 = 44\), so \(12b = 60\) and \(b = 5\). 6. The original sheet is \(5\,\text{in.}\) by \(10\,\text{in.}\). The base is \(1\,\text{in.}\) by \(6\,\text{in.}\), and the height is \(2\,\text{in.}\). 7. The volume is \(1 \cdot 6 \cdot 2 = 12\,\text{in.}^3\).

The original sheet is \(5\,\text{in.}\) by \(10\,\text{in.}\), and the box has volume \(12\,\text{in.}^3\).