# 8.3 Multiplying and dividing rational expressions  (Page 2/2)

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$\begin{array}{l}\left(x-3\right)·\frac{4x-9}{{x}^{2}-6x+9}.\\ \frac{\overline{)\left(x-3\right)}}{1}·\frac{4x-9}{\overline{)\left(x-3\right)}\left(x-3\right)}\\ \frac{4x-9}{x-3}\end{array}$

$\begin{array}{l}\begin{array}{lll}\frac{-{x}^{2}-3x-2}{{x}^{2}+8x+15}·\frac{4x+20}{{x}^{2}+2x}.\hfill & \hfill & \text{Factor\hspace{0.17em}–1 from the first numerator}.\hfill \\ \frac{-\left({x}^{2}+3x+2\right)}{{x}^{2}+8x+15}·\frac{4x+20}{{x}^{2}+2x}\hfill & \hfill & \text{Factor}.\hfill \\ \frac{-\left(x+1\right)\overline{)\left(x+2\right)}}{\left(x+3\right)\overline{)\left(x+5\right)}}·\frac{4\overline{)\left(x+5\right)}}{x\overline{)\left(x+2\right)}}\hfill & \hfill & \text{Multiply}\text{.}\hfill \end{array}\\ \begin{array}{ccc}\frac{-4\left(x+1\right)}{x\left(x+3\right)}=\frac{-4x-1}{x\left(x+3\right)}& \text{or}& \frac{-4x-1}{{x}^{2}+3x}\end{array}\end{array}$

## Practice set a

Perform each multiplication.

$\frac{5}{3}·\frac{6}{7}$

$\frac{10}{7}$

$\frac{{a}^{3}}{{b}^{2}{c}^{2}}·\frac{{c}^{5}}{{a}^{5}}$

$\frac{{c}^{3}}{{a}^{2}{b}^{2}}$

$\frac{y-1}{{y}^{2}+1}·\frac{y+1}{{y}^{2}-1}$

$\frac{1}{{y}^{2}+1}$

$\frac{{x}^{2}-x-12}{{x}^{2}+7x+6}·\frac{{x}^{2}-4x-5}{{x}^{2}-9x+20}$

$\frac{x+3}{x+6}$

$\frac{{x}^{2}+6x+8}{{x}^{2}-6x+8}·\frac{{x}^{2}-2x-8}{{x}^{2}+2x-8}$

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

## Division of rational expressions

To divide one rational expression by another, we first invert the divisor then multiply the two expressions. Symbolically, if we let $P,Q,R,$ and $S$ represent polynomials, we can write

$\frac{P}{Q}÷\frac{R}{S}=\frac{P}{Q}·\frac{S}{R}=\frac{P·S}{Q·R}$

## Sample set b

Perform the following divisions.

$\begin{array}{lll}\frac{6{x}^{2}}{5a}÷\frac{2x}{10{a}^{3}}.\hfill & \hfill & \text{Invert the divisor and multiply}\text{.}\hfill \\ \frac{\stackrel{3}{\overline{)6}}{x}^{\overline{)2}}}{\overline{)5}\overline{)a}}·\frac{\stackrel{2}{\overline{)10}}{a}^{\stackrel{2}{\overline{)3}}}}{\overline{)2}\overline{)x}}=\frac{3x·2{a}^{2}}{1}=6{a}^{2}x\hfill & \hfill & \hfill \end{array}$

$\begin{array}{lll}\frac{{x}^{2}+3x-10}{2x-2}÷\frac{{x}^{2}+9x+20}{{x}^{2}+3x-4}\hfill & \hfill & \text{Invert and multiply}\text{.}\hfill \\ \frac{{x}^{2}+3x-10}{2x-2}·\frac{{x}^{2}+3x-4}{{x}^{2}+9x+20}\hfill & \hfill & \text{Factor}\text{.}\hfill \\ \frac{\overline{)\left(x+5\right)}\left(x-2\right)}{2\overline{)\left(x-1\right)}}·\frac{\overline{)\left(x+4\right)}\overline{)\left(x-1\right)}}{\overline{)\left(x+5\right)}\overline{)\left(x+4\right)}}\hfill & \hfill & \hfill \\ \frac{x-2}{2}\hfill & \hfill & \hfill \end{array}$

$\begin{array}{lll}\left(4x+7\right)÷\frac{12x+21}{x-2}.\hfill & \hfill & \text{Write\hspace{0.17em}}\text{}4x+7\text{}\text{\hspace{0.17em}as\hspace{0.17em}}\text{}\frac{4x+7}{1}.\hfill \\ \frac{4x+7}{1}÷\frac{12x+21}{x-2}\hfill & \hfill & \text{Invert and multiply}\text{.}\hfill \\ \frac{4x+7}{1}·\frac{x-2}{12x+21}\hfill & \hfill & \text{Factor}\text{.}\hfill \\ \frac{\overline{)4x+7}}{1}·\frac{x-2}{3\left(\overline{)4x+7}\right)}=\frac{x-2}{3}\hfill & \hfill & \hfill \end{array}$

## Practice set b

Perform each division.

$\frac{8{m}^{2}n}{3{a}^{5}{b}^{2}}÷\frac{2m}{15{a}^{7}{b}^{2}}$

$20{a}^{2}mn$

$\frac{{x}^{2}-4}{{x}^{2}+x-6}÷\frac{{x}^{2}+x-2}{{x}^{2}+4x+3}$

$\frac{x+1}{x-1}$

$\frac{6{a}^{2}+17a+12}{3a+2}÷\left(2a+3\right)$

$\frac{3a+4}{3a+2}$

## Excercises

For the following problems, perform the multiplications and divisions.

$\frac{4{a}^{3}}{5b}·\frac{3b}{2a}$

$\frac{6{a}^{2}}{5}$

$\frac{9{x}^{4}}{4{y}^{3}}·\frac{10y}{{x}^{2}}$

$\frac{a}{b}·\frac{b}{a}$

1

$\frac{2x}{5y}·\frac{5y}{2x}$

$\frac{12{a}^{3}}{7}·\frac{28}{15a}$

$\frac{16{a}^{2}}{5}$

$\frac{39{m}^{4}}{16}·\frac{4}{13{m}^{2}}$

$\frac{18{x}^{6}}{7}·\frac{1}{4{x}^{2}}$

$\frac{9{x}^{4}}{14}$

$\frac{34{a}^{6}}{21}·\frac{42}{17{a}^{5}}$

$\frac{16{x}^{6}{y}^{3}}{15{x}^{2}}·\frac{25x}{4y}$

$\frac{20{x}^{5}{y}^{2}}{3}$

$\frac{27{a}^{7}{b}^{4}}{39b}·\frac{13{a}^{4}{b}^{2}}{16{a}^{5}}$

$\frac{10{x}^{2}{y}^{3}}{7{y}^{5}}·\frac{49y}{15{x}^{6}}$

$\frac{14}{3{x}^{4}y}$

$\frac{22{m}^{3}{n}^{4}}{11{m}^{6}n}·\frac{33mn}{4m{n}^{3}}$

$\frac{-10{p}^{2}q}{7{a}^{3}{b}^{2}}·\frac{21{a}^{5}{b}^{3}}{2p}$

$-15{a}^{2}bpq$

$\frac{-25{m}^{4}{n}^{3}}{14{r}^{3}{s}^{3}}·\frac{21r{s}^{4}}{10mn}$

$\frac{9}{a}÷\frac{3}{{a}^{2}}$

$3a$

$\frac{10}{{b}^{2}}÷\frac{4}{{b}^{3}}$

$\frac{21{a}^{4}}{5{b}^{2}}÷\frac{14a}{15{b}^{3}}$

$\frac{9{a}^{3}b}{2}$

$\frac{42{x}^{5}}{16{y}^{4}}÷\frac{21{x}^{4}}{8{y}^{3}}$

$\frac{39{x}^{2}{y}^{2}}{55{p}^{2}}÷\frac{13{x}^{3}y}{15{p}^{6}}$

$\frac{9{p}^{4}y}{11x}$

$\frac{14m{n}^{3}}{25{n}^{6}}÷\frac{32m}{20{m}^{2}{n}^{3}}$

$\frac{12{a}^{2}{b}^{3}}{-5x{y}^{4}}÷\frac{6{a}^{2}}{15{x}^{2}}$

$\frac{-6{b}^{3}x}{{y}^{4}}$

$\frac{24{p}^{3}q}{9m{n}^{3}}÷\frac{10pq}{-21{n}^{2}}$

$\frac{x+8}{x+1}·\frac{x+2}{x+8}$

$\frac{x+2}{x+1}$

$\frac{x+10}{x-4}·\frac{x-4}{x-1}$

$\frac{2x+5}{x+8}·\frac{x+8}{x-2}$

$\frac{2x+5}{x-2}$

$\frac{y+2}{2y-1}·\frac{2y-1}{y-2}$

$\frac{x-5}{x-1}÷\frac{x-5}{4}$

$\frac{4}{x-1}$

$\frac{x}{x-4}÷\frac{2x}{5x+1}$

$\frac{a+2b}{a-1}÷\frac{4a+8b}{3a-3}$

$\frac{3}{4}$

$\frac{6m+2}{m-1}÷\frac{4m-4}{m-1}$

${x}^{3}·\frac{4ab}{x}$

$4ab{x}^{2}$

${y}^{4}·\frac{3{x}^{2}}{{y}^{2}}$

$2{a}^{5}÷\frac{6{a}^{2}}{4b}$

$\frac{4{a}^{3}b}{3}$

$16{x}^{2}{y}^{3}÷\frac{10xy}{3}$

$21{m}^{4}{n}^{2}÷\frac{3m{n}^{2}}{7n}$

$49{m}^{3}n$

$\left(x+8\right)·\frac{x+2}{x+8}$

$\left(x-2\right)·\frac{x-1}{x-2}$

$x-1$

${\left(a-6\right)}^{3}·\frac{{\left(a+2\right)}^{2}}{a-6}$

${\left(b+1\right)}^{4}·\frac{{\left(b-7\right)}^{3}}{b+1}$

${\left(b+1\right)}^{3}{\left(b-7\right)}^{3}$

${\left({b}^{2}+2\right)}^{3}·\frac{b-3}{{\left({b}^{2}+2\right)}^{2}}$

${\left({x}^{3}-7\right)}^{4}·\frac{{x}^{2}-1}{{\left({x}^{3}-7\right)}^{2}}$

${\left({x}^{3}-7\right)}^{2}\left(x+1\right)\left(x-1\right)$

$\left(x-5\right)÷\frac{x-5}{x-2}$

$\left(y-2\right)÷\frac{y-2}{y-1}$

$\left(y-1\right)$

${\left(y+6\right)}^{3}÷\frac{{\left(y+6\right)}^{2}}{y-6}$

${\left(a-2b\right)}^{4}÷\frac{{\left(a-2b\right)}^{2}}{a+b}$

${\left(a-2b\right)}^{2}\left(a+b\right)$

$\frac{{x}^{2}+3x+2}{{x}^{2}-4x+3}·\frac{{x}^{2}-2x-3}{2x+2}$

$\frac{6x-42}{{x}^{2}-2x-3}·\frac{{x}^{2}-1}{x-7}$

$\frac{6\left(x-1\right)}{\left(x-3\right)}$

$\frac{3a+3b}{{a}^{2}-4a-5}÷\frac{9a+9b}{{a}^{2}-3a-10}$

$\frac{{a}^{2}-4a-12}{{a}^{2}-9}÷\frac{{a}^{2}-5a-6}{{a}^{2}+6a+9}$

$\frac{\left(a+2\right)\left(a+3\right)}{\left(a-3\right)\left(a+1\right)}$

$\frac{{b}^{2}-5b+6}{{b}^{2}-b-2}·\frac{{b}^{2}-2b-3}{{b}^{2}-9b+20}$

$\frac{{m}^{2}-4m+3}{{m}^{2}+5m-6}·\frac{{m}^{2}+4m-12}{{m}^{2}-5m+6}$

1

$\frac{{r}^{2}+7r+10}{{r}^{2}-2r-8}÷\frac{{r}^{2}+6r+5}{{r}^{2}-3r-4}$

$\frac{2{a}^{2}+7a+3}{3{a}^{2}-5a-2}·\frac{{a}^{2}-5a+6}{{a}^{2}+2a-3}$

$\frac{\left(2a+1\right)\left(a-6\right)\left(a+1\right)}{\left(3a+1\right)\left(a-1\right)\left(a-2\right)}$

$\frac{6{x}^{2}+x-2}{2{x}^{2}+7x-4}·\frac{{x}^{2}+2x-12}{3{x}^{2}-4x-4}$

$\frac{{x}^{3}y-{x}^{2}{y}^{2}}{{x}^{2}y-{y}^{2}}·\frac{{x}^{2}-y}{x-xy}$

$\frac{x\left(x-y\right)}{1-y}$

$\frac{4{a}^{3}b-4{a}^{2}{b}^{2}}{15a-10}·\frac{3a-2}{4ab-2{b}^{2}}$

$\frac{x+3}{x-4}·\frac{x-4}{x+1}·\frac{x-2}{x+3}$

$\frac{x-2}{x+1}$

$\frac{x-7}{x+8}·\frac{x+1}{x-7}·\frac{x+8}{x-2}$

$\frac{2a-b}{a+b}·\frac{a+3b}{a-5b}·\frac{a-5b}{2a-b}$

$\frac{a+3b}{a+b}$

$\frac{3a{\left(a+1\right)}^{2}}{a-5}·\frac{6{\left(a-5\right)}^{2}}{5a+5}·\frac{15a+30}{4a-20}$

$\frac{-3{a}^{2}}{4b}·\frac{-8{b}^{3}}{15a}$

$\frac{2a{b}^{2}}{5}$

$\frac{-6{x}^{3}}{5{y}^{2}}·\frac{20y}{-2x}$

$\frac{-8{x}^{2}{y}^{3}}{-5x}÷\frac{4}{-15xy}$

$-6{x}^{2}{y}^{4}$

$\frac{-4{a}^{3}}{3b}÷\frac{2a}{6{b}^{2}}$

$\frac{-3a-3}{2a+2}·\frac{{a}^{2}-3a+2}{{a}^{2}-5a-6}$

$\frac{-3\left(a-2\right)\left(a-1\right)}{2\left(a-6\right)\left(a+1\right)}$

$\frac{{x}^{2}-x-2}{{x}^{2}-3x-4}·\frac{-{x}^{2}+2x+3}{-4x-8}$

$\frac{-5x-10}{{x}^{2}-4x+3}·\frac{{x}^{2}+4x+1}{{x}^{2}+x-2}$

$\frac{-5\left({x}^{2}+4x+1\right)}{\left(x-3\right){\left(x-1\right)}^{2}}$

$\frac{-{a}^{2}-2a+15}{-6a-12}÷\frac{{a}^{2}-2a-8}{-2a-10}$

$\frac{-{b}^{2}-5b+14}{3b-6}÷\frac{-{b}^{2}-9b-14}{-b+8}$

$\frac{-\left(b-8\right)}{3\left(b+2\right)}$

$\frac{3a+6}{4a-24}·\frac{6-a}{3a+15}$

$\frac{4x+12}{x-7}·\frac{7-x}{2x+2}$

$\frac{-2\left(x+3\right)}{\left(x+1\right)}$

$\frac{-2b-2}{{b}^{2}+b-6}·\frac{-b+2}{b+5}$

$\frac{3{x}^{2}-6x-9}{2{x}^{2}-6x-4}÷\frac{3{x}^{2}-5x-2}{6{x}^{2}-7x-3}$

$\frac{3\left(x-3\right)\left(x+1\right)\left(2x-3\right)}{2\left({x}^{2}-3x-2\right)\left(x-2\right)}$

$\frac{-2{b}^{2}-2b+4}{8{b}^{2}-28b-16}÷\frac{{b}^{2}-2b+1}{2{b}^{2}-5b-3}$

$\frac{{x}^{2}+4x+3}{{x}^{2}+5x+4}÷\left(x+3\right)$

$\frac{\left(x+4\right)\left(x-1\right)}{\left(x+3\right)\left({x}^{2}-4x-3\right)}$

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

$\frac{3{x}^{2}-21x+18}{{x}^{2}+5x+6}÷\left(x+2\right)$

$\frac{3\left(x-6\right)\left(x-1\right)}{{\left(x+2\right)}^{2}\left(x+3\right)}$

## Exercises for review

( [link] ) If $a<0$ , then $|a|=$ .

( [link] ) Classify the polynomial $4xy+2y$ as a monomial, binomial, or trinomial. State its degree and write the numerical coefficient of each term.

binomial; 2; 4, 2

( [link] ) Find the product: ${y}^{2}\left(2y-1\right)\left(2y+1\right)$ .

( [link] ) Translate the sentence “four less than twice some number is two more than the number” into an equation.

$2x-4=x+2$

( [link] ) Reduce the fraction $\frac{{x}^{2}-4x+4}{{x}^{2}-4}$ .

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