5.4 Dividing Polynomials

Before you get started, take this readiness quiz.

Add: $\frac{3}{d} + \frac{x}{d} .$
If you missed this problem, review the reference.

Simplify: $\frac{30 x y^{3}}{5 x y} .$
If you missed this problem, review the reference.

Combine like terms: $8 a^{2} + 12 a + 1 + 3 a^{2} - 5 a + 4 .$
If you missed this problem, review the reference.

Find the quotient: $54 a^{2} b^{3} \div \left(−6 a b^{5}\right) .$

Find the quotient: $−72 a^{7} b^{3} \div \left(8 a^{12} b^{4}\right) .$

Find the quotient: $−63 c^{8} d^{3} \div \left(7 c^{12} d^{2}\right) .$

Find the quotient: $\frac{14 x^{7} y^{12}}{21 x^{11} y^{6}} .$

Find the quotient: $\frac{28 x^{5} y^{14}}{49 x^{9} y^{12}} .$

Find the quotient: $\frac{30 m^{5} n^{11}}{48 m^{10} n^{14}} .$

Find the quotient: $\left(18 x^{3} y - 36 x y^{2}\right) \div \left(−3 x y\right) .$

Find the quotient: $\left(32 a^{2} b - 16 a b^{2}\right) \div \left(−8 a b\right) .$

Find the quotient: $\left(−48 a^{8} b^{4} - 36 a^{6} b^{5}\right) \div \left(−6 a^{3} b^{3}\right) .$

Find the quotient: $\left(y^{2} + 10 y + 21\right) \div \left(y + 3\right) .$

Find the quotient: $\left(m^{2} + 9 m + 20\right) \div \left(m + 4\right) .$

Find the quotient: $\left(x^{4} - 7 x^{2} + 7 x + 6\right) \div \left(x + 3\right) .$

Find the quotient: $\left(x^{4} - 11 x^{2} - 7 x - 6\right) \div \left(x + 3\right) .$

Find the quotient: $\left(x^{3} - 64\right) \div \left(x - 4\right) .$

Find the quotient: $\left(125 x^{3} - 8\right) \div \left(5 x - 2\right) .$

Use synthetic division to find the quotient and remainder when $3 x^{3} + 10 x^{2} + 6 x - 2$ is divided by $x + 2 .$

Use synthetic division to find the quotient and remainder when $4 x^{3} + 5 x^{2} - 5 x + 3$ is divided by $x + 2 .$

Use synthetic division to find the quotient and remainder when $x^{4} - 16 x^{2} + 5 x + 20$ is divided by $x + 4 .$

Use synthetic division to find the quotient and remainder when $x^{4} - 9 x^{2} + 2 x + 6$ is divided by $x + 3 .$

For functions $f \left(x\right) = x^{2} - 5 x - 24$ and $g \left(x\right) = x + 3 ,$ find ⓐ $\left(\frac{f}{g}\right) \left(x\right)$ ⓑ $\left(\frac{f}{g}\right) \left(−3\right) .$

For functions $f \left(x\right) = x^{2} - 5 x - 36$ and $g \left(x\right) = x + 4 ,$ find ⓐ $\left(\frac{f}{g}\right) \left(x\right)$ ⓑ $\left(\frac{f}{g}\right) \left(−5\right) .$

Use the Remainder Theorem to find the remainder when $f \left(x\right) = x^{3} + 4 x + 15$ is divided by $x + 2 .$

Use the Remainder Theorem to find the remainder when $f \left(x\right) = x^{3} - 7 x + 12$ is divided by $x + 3 .$

Use the Remainder Theorem to determine if $x - 4$ is a factor of $f \left(x\right) = x^{3} - 64 .$

Use the Factor Theorem to determine if $x - 5$ is a factor of $f \left(x\right) = x^{3} - 125 .$

Use the Factor Theorem to determine if $x - 6$ is a factor of $f \left(x\right) = x^{3} - 216 .$

Divide the monomials.

$20 m^{8} n^{4} \div \left(30 m^{5} n^{9}\right)$

Divide the monomials.

$\frac{45 x^{5} y^{9}}{−60 x^{8} y^{6}}$

Divide the monomials.

$\frac{\left(−18 p^{4} q^{7}\right) \left(−6 p^{3} q^{8}\right)}{−36 p^{12} q^{10}}$

Divide the monomials.

$\frac{\left(4 u^{2} v^{5}\right) \left(15 u^{3} v\right)}{\left(12 u^{3} v\right) \left(u^{4} v\right)}$

Divide each polynomial by the monomial.

$\left(8 x^{3} + 6 x^{2}\right) \div 2 x$

Divide each polynomial by the monomial.

$\left(48 y^{4} - 24 y^{3}\right) \div \left(−8 y^{2}\right)$

Divide each polynomial by the monomial.

$\frac{72 r^{5} s^{2} + 132 r^{4} s^{3} - 96 r^{3} s^{5}}{12 r^{2} s^{2}}$

Divide each polynomial by the monomial.

$\frac{20 y^{2} + 12 y - 1}{−4 y}$

Divide each polynomial by the binomial.

$\left(a^{2} - 2 a - 35\right) \div \left(a + 5\right)$

Divide each polynomial by the binomial.

$\left(4 x^{2} - 17 x - 15\right) \div \left(x - 5\right)$

Divide each polynomial by the binomial.

$\left(p^{2} + 11 p + 16\right) \div \left(p + 8\right)$

Divide each polynomial by the binomial.

$\left(2 n^{3} - 10 n + 28\right) \div \left(n + 3\right)$

Divide each polynomial by the binomial.

$\left(m^{3} + 1000\right) \div \left(m + 10\right)$

Divide each polynomial by the binomial.

$\left(125 y^{3} - 64\right) \div \left(5 y - 4\right)$

Use synthetic Division to find the quotient and remainder.

$x^{3} - 3 x^{2} - 4 x + 12$ is divided by $x + 2$

Use synthetic Division to find the quotient and remainder.

$2 x^{3} - 11 x^{2} + 16 x - 12$ is divided by $x - 4$

Use synthetic Division to find the quotient and remainder.

$x^{4} + x^{2} + 6 x - 10$ is divided by $x + 2$

Use synthetic Division to find the quotient and remainder.

$3 x^{4} - 11 x^{3} + 2 x^{2} + 10 x + 6$ is divided by $x - 3$

Divide.

For functions $f \left(x\right) = x^{2} - 15 x + 54$ and $g \left(x\right) = x - 9 ,$ find ⓐ $\left(\frac{f}{g}\right) \left(x\right)$ ⓑ $\left(\frac{f}{g}\right) \left(−5\right)$

Divide.

For functions $f \left(x\right) = x^{3} + 2 x^{2} - 19 x + 12$ and $g \left(x\right) = x - 3 ,$ find ⓐ $\left(\frac{f}{g}\right) \left(x\right)$ ⓑ $\left(\frac{f}{g}\right) \left(0\right)$

Divide.

For functions $f \left(x\right) = x^{2} + 2 x - 3$ and $g \left(x\right) = x + 3$ find ⓐ $\left(\frac{f}{g}\right) \left(x\right)$ⓑ $\left(\frac{f}{g}\right) \left(3\right)$

Use the Remainder Theorem to find the remainder.

$f \left(x\right) = x^{3} - 4 x - 9$ is divided by $x + 2$

Use the Remainder Theorem to find the remainder.

$f \left(x\right) = 7 x^{2} - 5 x - 8$ divided by $x - 1$

Use the Factor Theorem to determine if $x - c$ is a factor of the polynomial function.

Determine whether $x + 4$ a factor of $x^{3} + x^{2} - 14 x + 8$

Use the Factor Theorem to determine if $x - c$ is a factor of the polynomial function.

Determine whether $x - 3$ a factor of $x^{3} - 7 x^{2} + 11 x + 3$

Divide $\frac{10 x^{2} + x - 12}{2 x}$ and explain with words how you get each term of the quotient.

In your own words, write the steps for synthetic division for $x^{2} + 5 x + 6$ divided by $x - 2 .$

Determine the type of polynomial.

$5 m + 9$

Determine the type of polynomial.

$y^{2} + 6 y^{3} + 9 y^{4}$

Add or subtract the polynomials.

$−8 y^{3} - 5 y^{3}$

Add or subtract the polynomials.

$\left(8 m^{2} + 12 m - 5\right) - \left(2 m^{2} - 7 m - 1\right)$

Add or subtract the polynomials.

$\left(5 u^{2} + 8 u\right) - \left(4 u - 7\right)$

Add or subtract the polynomials.

Find the difference of $x^{2} + 6 x + 8$ and $x^{2} - 8 x + 15 .$

Simplify.

$18 a - 7 b - 21 a$

Simplify.

$\left(6 a^{2} + 7\right) + \left(2 a^{2} - 5 a - 9\right)$

Simplify.

$\left(7 m^{2} - 2 m - 5\right) - \left(4 m^{2} + m - 8\right)$

Simplify.

Subtract $\left(8 y^{2} - y + 9\right)$ from $\left(11 y^{2} - 9 y - 5\right)$

Simplify.

$\left(x^{3} - x^{2} y\right) - \left(4 x y^{2} - y^{3}\right) + \left(3 x^{2} y - x y^{2}\right)$

Find the function values for each polynomial function.

For the function $f \left(x\right) = 7 x^{2} - 3 x + 5$ find:
ⓐ $f \left(5\right)$ ⓑ $f \left(−2\right)$ ⓒ $f \left(0\right)$

Find the function values for each polynomial function.

A pair of glasses is dropped off a bridge 640 feet above a river. The polynomial function $h \left(t\right) = −16 t^{2} + 640$ gives the height of the glasses t seconds after they were dropped. Find the height of the glasses when $t = 6 .$

Find ⓐ (f + g)(x) ⓑ (f + g)(3) ⓒ (f − g)(x) ⓓ (f − g)(−2)

$f \left(x\right) = 2 x^{2} - 4 x - 7$ and $g \left(x\right) = 2 x^{2} - x + 5$

Simplify each expression using the properties for exponents.

$p^{3} \cdot p^{10}$

Simplify each expression using the properties for exponents.

$a \cdot a^{2} \cdot a^{3}$

Simplify each expression using the properties for exponents.

$y^{a} \cdot y^{b}$

Simplify each expression using the properties for exponents.

$\frac{a^{6}}{a}$

Simplify each expression using the properties for exponents.

$\frac{1}{x^{5}}$

Simplify each expression using the properties for exponents.

$y^{0}$

Simplify each expression using the properties for exponents.

$12 a^{0} - 15 b^{0}$

Simplify each expression.

$\left(−10\right)^{−3}$

Simplify each expression.

$\left(8 n\right)^{−1}$

Simplify each expression.

$10^{−3}$

Simplify each expression.

$\frac{1}{6^{−2}}$

Simplify each expression.

$\left(- \frac{1}{5}\right)^{−3}$

Simplify each expression.

$\left(−5\right)^{−3}$

Simplify each expression.

$\left(- \frac{3}{x}\right)^{−3}$

Simplify each expression using the Product Property.

$\left(3^{2}\right)^{5}$

Simplify each expression using the Product Property.

$x^{−3} \cdot x^{9}$

Simplify each expression using the Product Property.

$\left(u v^{−3}\right) \left(u^{−4} v^{−2}\right)$

Simplify each expression using the Product Property.

$p^{5} \cdot p^{−2} \cdot p^{−4}$

Simplify each expression using the Power Property.

$\frac{q^{4}}{q^{20}}$

Simplify each expression using the Power Property.

$\frac{n^{−3}}{n^{−5}}$

Simplify each expression using the Product to a Power Property.

$\left(−4 p q\right)^{0}$

Simplify each expression using the Product to a Power Property.

$\left(3 y^{−4}\right)^{2}$

Simplify each expression using the Quotient to a Power Property.

$\left(\frac{3 x y^{2}}{z}\right)^{4}$

Simplify each expression by applying several properties.

$\left(x^{2} y\right)^{2} \left(3 x y^{5}\right)^{3}$

Simplify each expression by applying several properties.

$\left(\frac{3 x y^{3}}{4 x^{4} y^{−2}}\right)^{2} \left(\frac{6 x y^{4}}{8 x^{3} y^{−2}}\right)^{−1}$

Write each number in scientific notation.

5,300,000

Convert each number to decimal form.

$2.9 \times 10^{4}$

Convert each number to decimal form.

$9.413 \times 10^{−5}$

Multiply or divide as indicated. Write your answer in decimal form.

$\left(1.5 \times 10^{−3}\right) \left(4.8 \times 10^{−1}\right)$

Multiply or divide as indicated. Write your answer in decimal form.

$\frac{9 \times 10^{−3}}{1 \times 10^{−6}}$

Multiply the monomials.

$\left(\frac{1}{3} c^{2}\right) \left(30 c^{8}\right)$

Multiply the monomials.

$\left(\frac{2}{3} m^{3} n^{6}\right) \left(\frac{1}{6} m^{4} n^{4}\right)$

Multiply.

$a^{2} \left(a^{2} - 9 a - 36\right)$

Multiply.

$\left(4 n - 5\right) \left(2 n^{3}\right)$

ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical Method.

$\left(y - 4\right) \left(y + 12\right)$

ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical Method.

$\left(6 p - 11\right) \left(3 p - 10\right)$

Multiply the binomials. Use any method.

$\left(k + 6\right) \left(k - 9\right)$

Multiply the binomials. Use any method.

$\left(2 y - 9\right) \left(5 y - 7\right)$

Multiply the binomials. Use any method.

$\left(x - 8\right) \left(x + 9\right)$

Multiply the binomials. Use any method.

$\left(10 a - 1\right) \left(3 a - 3\right)$

Multiply using ⓐ the Distributive Property ⓑ the Vertical Method.

$\left(5 b - 2\right) \left(3 b^{2} + b - 9\right)$

Multiply. Use either method.

$\left(4 y - 1\right) \left(6 y^{2} - 12 y + 5\right)$

Square each binomial using the Binomial Squares Pattern.

$\left(x + \frac{3}{4}\right)^{2}$

Square each binomial using the Binomial Squares Pattern.

$\left(5 p + 7 q\right)^{2}$

Multiply each pair of conjugates using the Product of Conjugates.

$\left(6 x + y\right) \left(6 x - y\right)$

Multiply each pair of conjugates using the Product of Conjugates.

$\left(12 x^{3} - 7 y^{2}\right) \left(12 x^{3} + 7 y^{2}\right)$

Divide the monomials.

$72 p^{12} \div 8 p^{3}$

Divide the monomials.

$\frac{45 y^{6}}{−15 y^{10}}$

Divide the monomials.

$\frac{28 a^{9} b}{7 a^{4} b^{3}}$

Divide the monomials.

$\frac{\left(5 m^{9} n^{3}\right) \left(8 m^{3} n^{2}\right)}{\left(10 m n^{4}\right) \left(m^{2} n^{5}\right)}$

Divide each polynomial by the monomial

$\left(54 y^{4} - 24 y^{3}\right) \div \left(−6 y^{2}\right)$

Divide each polynomial by the monomial

$\frac{12 x^{2} + 4 x - 3}{−4 x}$

Divide each polynomial by the binomial.

$\left(y^{2} + 2 y + 18\right) \div \left(y + 5\right)$

Divide each polynomial by the binomial.

$\left(a^{3} - 1\right) \div \left(a + 1\right)$

In the following exercises, use synthetic Division to find the quotient and remainder.

$2 x^{3} - 11 x^{2} + 11 x + 12$ is divided by $x - 3$

Divide.

For functions $f \left(x\right) = x^{2} - 15 x + 54$ and $g \left(x\right) = x - 9 ,$ find ⓐ $\left(\frac{f}{g}\right) \left(x\right)$
ⓑ $\left(\frac{f}{g}\right) \left(−2\right)$

Use the Remainder Theorem to find the remainder.

$f \left(x\right) = x^{3} - 4 x - 9$ is divided by $x + 2$

Use the Factor Theorem to determine if $x - c$ is a factor of the polynomial function.

Determine whether $x - 2$ is a factor of $x^{3} - 7 x^{2} + 7 x - 6$.

For the polynomial $8 y^{4} - 3 y^{2} + 1$

ⓐ Is it a monomial, binomial, or trinomial? ⓑ What is its degree?

$\left(10 x^{2} - 3 x + 5\right) - \left(4 x^{2} - 6\right)$

$x^{−3} x^{4}$

$\left(47 a^{18} b^{23} c^{5}\right)^{0}$

$\left(2 y\right)^{−3}$

$\frac{x^{4}}{x^{−5}}$

$\frac{24 r^{3} s}{6 r^{2} s^{7}}$

$\left(8 x y^{3}\right) \left(−6 x^{4} y^{6}\right)$

$\left(m + 3\right) \left(7 m - 2\right)$

$\left(4 x - 3\right)^{2}$

$\left(15 x y^{3} - 35 x^{2} y\right) \div 5 x y$

Use the Factor Theorem to determine if $x + 3$ a factor of $x^{3} + 8 x^{2} + 21 x + 18 .$

Simplify and write your answer in exponential notation.

$\left(2.4 \times 10^{8}\right) \left(2 \times 10^{−5}\right)$

Simplify and write your answer in exponential notation.

For the function $f \left(x\right) = 6 x^{2} - 3 x - 9$ find:
ⓐ $f \left(3\right)$ ⓑ $f \left(−2\right)$ ⓒ $f \left(0\right)$

Simplify and write your answer in exponential notation.

For functions
$f \left(x\right) = 3 x^{2} - 23 x - 36$ and
$g \left(x\right) = x - 9 ,$ find
ⓐ $\left(\frac{f}{g}\right) \left(x\right)$ ⓑ $\left(\frac{f}{g}\right) \left(3\right)$