7.1 Multiply and Divide Rational Expressions

Before you get started, take this readiness quiz.

Simplify: $\frac{90 y}{15 y^{2}} .$
If you missed this problem, review the reference.

Multiply: $\frac{14}{15} \cdot \frac{6}{35} .$
If you missed this problem, review the reference.

Divide: $\frac{12}{10} \div \frac{8}{25} .$
If you missed this problem, review the reference.

Determine the value for which each rational expression is undefined:

ⓐ $\frac{8 a^{2} b}{3 c}$ ⓑ $\frac{4 b - 3}{2 b + 5}$ ⓒ $\frac{x + 4}{x^{2} + 5 x + 6} .$

Determine the value for which each rational expression is undefined.

ⓐ $\frac{3 y^{2}}{8 x}$ ⓑ $\frac{8 n - 5}{3 n + 1}$ ⓒ $\frac{a + 10}{a^{2} + 4 a + 3}$

Determine the value for which each rational expression is undefined.

ⓐ $\frac{4 p}{5 q}$ ⓑ $\frac{y - 1}{3 y + 2}$ ⓒ $\frac{m - 5}{m^{2} + m - 6}$

Simplify: $\frac{x^{2} - x - 2}{x^{2} - 3 x + 2} .$

Simplify: $\frac{x^{2} - 3 x - 10}{x^{2} + x - 2} .$

Simplify: $\frac{3 a^{2} - 12 a b + 12 b^{2}}{6 a^{2} - 24 b^{2}}$.

Simplify: $\frac{2 x^{2} - 12 x y + 18 y^{2}}{3 x^{2} - 27 y^{2}}$.

Simplify: $\frac{5 x^{2} - 30 x y + 25 y^{2}}{2 x^{2} - 50 y^{2}}$.

Simplify: $\frac{x^{2} - 4 x - 5}{25 - x^{2}} .$

Simplify: $\frac{x^{2} + x - 2}{1 - x^{2}} .$

Simplify: $\frac{5 x}{x^{2} + 5 x + 6} \cdot \frac{x^{2} - 4}{10 x} .$

Simplify: $\frac{9 x^{2}}{x^{2} + 11 x + 30} \cdot \frac{x^{2} - 36}{3 x^{2}} .$

Multiply: $\frac{3 a^{2} - 8 a - 3}{a^{2} - 25} \cdot \frac{a^{2} + 10 a + 25}{3 a^{2} - 14 a - 5} .$

Simplify: $\frac{2 x^{2} + 5 x - 12}{x^{2} - 16} \cdot \frac{x^{2} - 8 x + 16}{2 x^{2} - 13 x + 15} .$

Simplify: $\frac{4 b^{2} + 7 b - 2}{1 - b^{2}} \cdot \frac{b^{2} - 2 b + 1}{4 b^{2} + 15 b - 4} .$

Simplify: $\frac{x^{3} + 8}{3 x^{2} - 6 x + 12} \div \frac{x^{2} - 4}{6} .$

Simplify: $\frac{2 z^{2}}{z^{2} - 1} \div \frac{z^{3} - z^{2} + z}{z^{3} + 1} .$

Divide: $\frac{\frac{6 x^{2} - 7 x + 2}{4 x - 8}}{\frac{2 x^{2} - 7 x + 3}{x^{2} - 5 x + 6}} .$

Simplify: $\frac{\frac{3 x^{2} + 7 x + 2}{4 x + 24}}{\frac{3 x^{2} - 14 x - 5}{x^{2} + x - 30}} .$

Simplify: $\frac{\frac{y^{2} - 36}{2 y^{2} + 11 y - 6}}{\frac{2 y^{2} - 2 y - 60}{8 y - 4}} .$

Perform the indicated operations: $\frac{4 m + 4}{3 m - 15} \cdot \frac{m^{2} - 3 m - 10}{m^{2} - 4 m - 32} \div \frac{12 m - 36}{6 m - 48} .$

Perform the indicated operations: $\frac{2 n^{2} + 10 n}{n - 1} \div \frac{n^{2} + 10 n + 24}{n^{2} + 8 n - 9} \cdot \frac{n + 4}{8 n^{2} + 12 n} .$

Find the domain of $R \left(x\right) = \frac{2 x^{2} - 14 x}{4 x^{2} - 16 x - 48} .$

Find the domain of $R \left(x\right) = \frac{2 x^{2} - 10 x}{4 x^{2} - 16 x - 20} .$

Find the domain of $R \left(x\right) = \frac{4 x^{2} - 16 x}{8 x^{2} - 16 x - 64} .$

Find $R \left(x\right) = f \left(x\right) \cdot g \left(x\right)$ where $f \left(x\right) = \frac{2 x - 6}{x^{2} - 8 x + 15}$ and $g \left(x\right) = \frac{x^{2} - 25}{2 x + 10} .$

Find $R \left(x\right) = f \left(x\right) \cdot g \left(x\right)$ where $f \left(x\right) = \frac{3 x - 21}{x^{2} - 9 x + 14}$ and $g \left(x\right) = \frac{2 x^{2} - 8}{3 x + 6} .$

Find $R \left(x\right) = f \left(x\right) \cdot g \left(x\right)$ where $f \left(x\right) = \frac{x^{2} - x}{3 x^{2} + 27 x - 30}$ and $g \left(x\right) = \frac{x^{2} - 100}{x^{2} - 10 x} .$

Find $R \left(x\right) = \frac{f \left(x\right)}{g \left(x\right)}$ where $f \left(x\right) = \frac{3 x^{2}}{x^{2} - 4 x}$ and $g \left(x\right) = \frac{9 x^{2} - 45 x}{x^{2} - 7 x + 10} .$

Find $R \left(x\right) = \frac{f \left(x\right)}{g \left(x\right)}$ where $f \left(x\right) = \frac{2 x^{2}}{x^{2} - 8 x}$ and $g \left(x\right) = \frac{8 x^{2} + 24 x}{x^{2} + x - 6} .$

Find $R \left(x\right) = \frac{f \left(x\right)}{g \left(x\right)}$ where $f \left(x\right) = \frac{15 x^{2}}{3 x^{2} + 33 x}$ and $g \left(x\right) = \frac{5 x - 5}{x^{2} + 9 x - 22} .$

Determine the values for which the rational expression is undefined.

ⓐ $\frac{2 x^{2}}{z}$, ⓑ $\frac{4 p - 1}{6 p - 5}$, ⓒ $\frac{n - 3}{n^{2} + 2 n - 8}$

Determine the values for which the rational expression is undefined.

ⓐ $\frac{4 x^{2} y}{3 y}$, ⓑ $\frac{3 x - 2}{2 x + 1}$, ⓒ $\frac{u - 1}{u^{2} - 3 u - 28}$

Simplify each rational expression.

$- \frac{44}{55}$

Simplify each rational expression.

$\frac{8 m^{3} n}{12 m n^{2}}$

Simplify each rational expression.

$\frac{8 n - 96}{3 n - 36}$

Simplify each rational expression.

$\frac{x^{2} + 4 x - 5}{x^{2} - 2 x + 1}$

Simplify each rational expression.

$\frac{a^{2} - 4}{a^{2} + 6 a - 16}$

Simplify each rational expression.

$\frac{p^{3} + 3 p^{2} + 4 p + 12}{p^{2} + p - 6}$

Simplify each rational expression.

$\frac{8 b^{2} - 32 b}{2 b^{2} - 6 b - 80}$

Simplify each rational expression.

$\frac{3 m^{2} + 30 m n + 75 n^{2}}{4 m^{2} - 100 n^{2}}$

Simplify each rational expression.

$\frac{a - 5}{5 - a}$

Simplify each rational expression.

$\frac{20 - 5 y}{y^{2} - 16}$

Simplify each rational expression.

$\frac{w^{3} + 216}{w^{2} - 36}$

Simplify each rational expression.

$\frac{z^{2} - 9 z + 20}{16 - z^{2}}$

Multiply the rational expressions.

$\frac{12}{16} \cdot \frac{4}{10}$

Multiply the rational expressions.

$\frac{5 x^{2} y^{4}}{12 x y^{3}} \cdot \frac{6 x^{2}}{20 y^{2}}$

Multiply the rational expressions.

$\frac{5 p^{2}}{p^{2} - 5 p - 36} \cdot \frac{p^{2} - 16}{10 p}$

Multiply the rational expressions.

$\frac{2 y^{2} - 10 y}{y^{2} + 10 y + 25} \cdot \frac{y + 5}{6 y}$

Multiply the rational expressions.

$\frac{28 - 4 b}{3 b - 3} \cdot \frac{b^{2} + 8 b - 9}{b^{2} - 49}$

Multiply the rational expressions.

$\frac{3 c^{2} - 16 c + 5}{c^{2} - 25} \cdot \frac{c^{2} + 10 c + 25}{3 c^{2} - 14 c - 5}$

Multiply the rational expressions.

$\frac{6 m^{2} - 13 m + 2}{9 - m^{2}} \cdot \frac{m^{2} - 6 m + 9}{6 m^{2} + 23 m - 4}$

Divide the rational expressions.

$\frac{v - 5}{11 - v} \div \frac{v^{2} - 25}{v - 11}$

Divide the rational expressions.

$\frac{3 s^{2}}{s^{2} - 16} \div \frac{s^{3} + 4 s^{2} + 16 s}{s^{3} - 64}$

Divide the rational expressions.

$\frac{p^{3} + q^{3}}{3 p^{2} + 3 p q + 3 q^{2}} \div \frac{p^{2} - q^{2}}{12}$

Divide the rational expressions.

$\frac{x^{2} + 3 x - 10}{4 x} \div \left(2 x^{2} + 20 x + 50\right)$

Divide the rational expressions.

$\frac{\frac{2 a^{2} - a - 21}{5 a + 20}}{\frac{a^{2} + 7 a + 12}{a^{2} + 8 a + 16}}$

Divide the rational expressions.

$\frac{\frac{12 c^{2} - 12}{2 c^{2} - 3 c + 1}}{\frac{4 c + 4}{6 c^{2} - 13 c + 5}}$

Perform the indicated operations.

$\frac{10 m^{2} + 80 m}{3 m - 9} \cdot \frac{m^{2} + 4 m - 21}{m^{2} - 9 m + 20} \div \frac{5 m^{2} + 10 m}{2 m - 10}$

Perform the indicated operations.

$\frac{12 p^{2} + 3 p}{p + 3} \div \frac{p^{2} + 2 p - 63}{p^{2} - p - 12} \cdot \frac{p - 7}{9 p^{3} - 9 p^{2}}$

Find the domain of each function.

$R \left(x\right) = \frac{x^{3} - 2 x^{2} - 25 x + 50}{x^{2} - 25}$

Find the domain of each function.

$R \left(x\right) = \frac{3 x^{2} + 15 x}{6 x^{2} + 6 x - 36}$

Find $R \left(x\right) = f \left(x\right) \cdot g \left(x\right)$ where $f \left(x\right)$ and $g \left(x\right)$ are given.

$f \left(x\right) = \frac{6 x^{2} - 12 x}{x^{2} + 7 x - 18}$
$g \left(x\right) = \frac{x^{2} - 81}{3 x^{2} - 27 x}$

Find $R \left(x\right) = f \left(x\right) \cdot g \left(x\right)$ where $f \left(x\right)$ and $g \left(x\right)$ are given.

$f \left(x\right) = \frac{4 x}{x^{2} - 3 x - 10}$
$g \left(x\right) = \frac{x^{2} - 25}{8 x^{2}}$

Find $R \left(x\right) = \frac{f \left(x\right)}{g \left(x\right)}$ where $f \left(x\right)$ and $g \left(x\right)$ are given.

$f \left(x\right) = \frac{27 x^{2}}{3 x - 21}$
$g \left(x\right) = \frac{3 x^{2} + 18 x}{x^{2} + 13 x + 42}$

Find $R \left(x\right) = \frac{f \left(x\right)}{g \left(x\right)}$ where $f \left(x\right)$ and $g \left(x\right)$ are given.

$f \left(x\right) = \frac{16 x^{2}}{4 x + 36}$
$g \left(x\right) = \frac{4 x^{2} - 24 x}{x^{2} + 4 x - 45}$

Explain how you find the values of x for which the rational expression $\frac{x^{2} - x - 20}{x^{2} - 4}$ is undefined.

ⓐ Multiply $\frac{7}{4} \cdot \frac{9}{10}$ and explain all your steps. ⓑ Multiply $\frac{n}{n - 3} \cdot \frac{9}{n + 3}$ and explain all your steps. ⓒ Evaluate your answer to part ⓑ when $n = 7$. Did you get the same answer you got in part ⓐ ? Why or why not?