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Divide: $\frac{\frac{6{x}^{2}-7x+2}{4x-8}}{\frac{2{x}^{2}-7x+3}{{x}^{2}-5x+6}}.$
$\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{\frac{6{x}^{2}-7x+2}{4x-8}}{\frac{2{x}^{2}-7x+3}{{x}^{2}-5x+6}}\hfill \\ \\ \\ \text{Rewrite with a division sign.}\hfill & & & \phantom{\rule{5em}{0ex}}\frac{6{x}^{2}-7x+2}{4x-8}\xf7\frac{2{x}^{2}-7x+3}{{x}^{2}-5x+6}\hfill \\ \\ \\ \begin{array}{c}\text{Rewrite as product of first times}\hfill \\ \text{reciprocal of second.}\hfill \end{array}\hfill & & & \phantom{\rule{5em}{0ex}}\frac{6{x}^{2}-7x+2}{4x-8}\xb7\frac{{x}^{2}-5x+6}{2{x}^{2}-7x+3}\hfill \\ \\ \\ \begin{array}{c}\text{Factor the numerators and the}\hfill \\ \text{denominators, and then multiply.}\hfill \end{array}\hfill & & & \phantom{\rule{5em}{0ex}}\frac{\left(2x-1\right)\left(3x-2\right)\left(x-2\right)\left(x-3\right)}{4\left(x-2\right)\left(2x-1\right)\left(x-3\right)}\hfill \\ \\ \\ \text{Simplify by dividing out common factors.}\hfill & & & \phantom{\rule{5em}{0ex}}\frac{\overline{)\left(2x-1\right)}\left(3x-2\right)\overline{)\left(x-2\right)}\overline{)\left(x-3\right)}}{4\overline{)\left(x-2\right)}\overline{)\left(2x-1\right)}\overline{)\left(x-3\right)}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{3x-2}{4}\hfill \end{array}$
Divide: $\frac{\frac{3{x}^{2}+7x+2}{4x+24}}{\frac{3{x}^{2}-14x-5}{{x}^{2}+x-30}}.$
$\frac{x+2}{4}$
Divide: $\frac{\frac{{y}^{2}-36}{2{y}^{2}+11y-6}}{\frac{2{y}^{2}-2y-60}{8y-4}}.$
$\frac{2}{y+5}$
If we have more than two rational expressions to work with, we still follow the same procedure. The first step will be to rewrite any division as multiplication by the reciprocal. Then we factor and multiply.
Divide: $\frac{3x-6}{4x-4}\xb7\frac{{x}^{2}+2x-3}{{x}^{2}-3x-10}\xf7\frac{2x+12}{8x+16}.$
Rewrite the division as multiplication by the reciprocal. | |
Factor the numerators and the denominators, and then multiply. | |
Simplify by dividing out common factors. | |
Simplify. |
Divide: $\frac{4m+4}{3m-15}\xb7\frac{{m}^{2}-3m-10}{{m}^{2}-4m-32}\xf7\frac{12m-36}{6m-48}.$
$\frac{2(m+1)(m+2)}{3(m+4)(m-3)}$
Divide: $\frac{2{n}^{2}+10n}{n-1}\xf7\frac{{n}^{2}+10n+24}{{n}^{2}+8n-9}\xb7\frac{n+4}{8{n}^{2}+12n}.$
$\frac{(n+5)(n+9)}{2(n+6)(2n+3)}$
Multiply Rational Expressions
In the following exercises, multiply.
$\frac{32}{5}\xb7\frac{16}{24}$
$\frac{21}{36}\xb7\frac{45}{24}$
$\frac{5{x}^{2}{y}^{4}}{12x{y}^{3}}\xb7\frac{6{x}^{2}}{20{y}^{2}}$
$\frac{{x}^{3}}{8y}$
$\frac{8{w}^{3}y}{9{y}^{2}}\xb7\frac{3y}{4{w}^{4}}$
$\frac{12{a}^{3}b}{{b}^{2}}\xb7\frac{2a{b}^{2}}{9{b}^{3}}$
$\frac{8ab}{3}$
$\frac{4m{n}^{2}}{5{n}^{3}}\xb7\frac{m{n}^{3}}{8{m}^{2}{n}^{2}}$
$\frac{5{p}^{2}}{{p}^{2}-5p-36}\xb7\frac{{p}^{2}-16}{10p}$
$\frac{p(p-4)}{2(p-9)}$
$\frac{3{q}^{2}}{{q}^{2}+q-6}\xb7\frac{{q}^{2}-9}{9q}$
$\frac{4r}{{r}^{2}-3r-10}\xb7\frac{{r}^{2}-25}{8{r}^{2}}$
$\frac{r+5}{2r(r+2)}$
$\frac{s}{{s}^{2}-9s+14}\xb7\frac{{s}^{2}-49}{7{s}^{2}}$
$\frac{{x}^{2}-7x}{{x}^{2}+6x+9}\xb7\frac{x+3}{4x}$
$\frac{x-7}{4(x+3)}$
$\frac{2{y}^{2}-10y}{{y}^{2}+10y+25}\xb7\frac{y+5}{6y}$
$\frac{{z}^{2}+3z}{{z}^{2}-3z-4}\xb7\frac{z-4}{{z}^{2}}$
$\frac{z+3}{z(z+1)}$
$\frac{2{a}^{2}+8a}{{a}^{2}-9a+20}\xb7\frac{a-5}{{a}^{2}}$
$\frac{28-4b}{3b-3}\xb7\frac{{b}^{2}+8b-9}{{b}^{2}-49}$
$-\frac{4(b+9)}{3(b+7)}$
$\frac{18c-2{c}^{2}}{6c+30}\xb7\frac{{c}^{2}+7c+10}{{c}^{2}-81}$
$\frac{35d-7{d}^{2}}{{d}^{2}+7d}\xb7\frac{{d}^{2}+12d+35}{{d}^{2}-25}$
$\mathrm{-7}$
$\frac{72m-12{m}^{2}}{8m+32}\xb7\frac{{m}^{2}+10m+24}{{m}^{2}-36}$
$\frac{4n+20}{{n}^{2}+n-20}\xb7\frac{{n}^{2}-16}{4n+16}$
$1$
$\frac{6{p}^{2}-6p}{{p}^{2}+7p-18}\xb7\frac{{p}^{2}-81}{3{p}^{2}-27p}$
$\frac{{q}^{2}-2q}{{q}^{2}+6q-16}\xb7\frac{{q}^{2}-64}{{q}^{2}-8q}$
$1$
$\frac{2{r}^{2}-2r}{{r}^{2}+4r-5}\xb7\frac{{r}^{2}-25}{2{r}^{2}-10r}$
Divide Rational Expressions
In the following exercises, divide.
$\frac{t-6}{3-t}\xf7\frac{{t}^{2}-9}{t-5}$
$-\frac{2t}{{t}^{3}-5t-9}$
$\frac{v-5}{11-v}\xf7\frac{{v}^{2}-25}{v-11}$
$\frac{10+w}{w-8}\xf7\frac{100-{w}^{2}}{8-w}$
$-\frac{1}{10-w}$
$\frac{7+x}{x-6}\xf7{\frac{49-x}{x+6}}^{2}$
$\frac{27{y}^{2}}{3y-21}\xf7\frac{3{y}^{2}+18}{{y}^{2}+13y+42}$
$\frac{3{y}^{2}(y+6)(y+7)}{(y-7)({y}^{2}+6)}$
$\frac{24{z}^{2}}{2z-8}\xf7\frac{4z-28}{{z}^{2}-11z+28}$
$\frac{16{a}^{2}}{4a+36}\xf7\frac{4{a}^{2}-24a}{{a}^{2}+4a-45}$
$\frac{a(a-5)}{a-6}$
$\frac{24{b}^{2}}{2b-4}\xf7\frac{12{b}^{2}+36b}{{b}^{2}-11b+18}$
$\frac{5{c}^{2}+9c+2}{{c}^{2}-25}\xf7\frac{3{c}^{2}-14c-5}{{c}^{2}+10c+25}$
$\frac{(c+2)(c+2)}{(c-2)(c-3)}$
$\frac{2{d}^{2}+d-3}{{d}^{2}-16}\xf7\frac{2{d}^{2}-9d-18}{{d}^{2}-8d+16}$
$\frac{6{m}^{2}-2m-10}{9-{m}^{2}}\xf7\frac{6{m}^{2}+29m-20}{{m}^{2}-6m+9}$
$-\frac{(m-2)(m-3)}{(3+m)(m+4)}$
$\frac{2{n}^{2}-3n-14}{25-{n}^{2}}\xf7\frac{2{n}^{2}-13n+21}{{n}^{2}-10n+25}$
$\frac{3{s}^{2}}{{s}^{2}-16}\xf7\frac{{s}^{3}-4{s}^{2}+16s}{{s}^{3}-64}$
$\frac{3s}{s+4}$
$\frac{{r}^{2}-9}{15}\xf7\frac{{r}^{3}-27}{5{r}^{2}+15r+45}$
$\frac{{p}^{3}+{q}^{3}}{3{p}^{2}+3pq+3{q}^{2}}\xf7\frac{{p}^{2}-{q}^{2}}{12}$
$\frac{4({p}^{2}-pq+{q}^{2})}{(p-q)({p}^{2}+pq+{q}^{2})}$
$\frac{{v}^{3}-8{w}^{3}}{2{v}^{2}+4vw+8{w}^{2}}\xf7\frac{{v}^{2}-4{w}^{2}}{4}$
$\frac{{t}^{2}-9}{2t}\xf7({t}^{2}-6t+9)$
$\frac{t+3}{2t(t-3)}$
$\frac{{x}^{2}+3x-10}{4x}\xf7(2{x}^{2}+20x+50)$
$\frac{2{y}^{2}-10yz-48{z}^{2}}{2y-1}\xf7(4{y}^{2}-32yz)$
$\frac{y+3z}{2y(2y-1)}$
$\frac{2{m}^{2}-98{n}^{2}}{2m+6}\xf7({m}^{2}-7mn)$
$\frac{\frac{2{a}^{2}-a-21}{5a+20}}{\frac{{a}^{2}+7a+12}{{a}^{2}+8a+16}}$
$\frac{2a-7}{5}$
$\frac{\frac{3{b}^{2}+2b-8}{12b+18}}{\frac{3{b}^{2}+2b-8}{2{b}^{2}-7b-15}}$
$\frac{\frac{12{c}^{2}-12}{2{c}^{2}-3c+1}}{\frac{4c+4}{6{c}^{2}-13c+5}}$
$3\left(3c-5\right)$
$\frac{\frac{4{d}^{2}+7d-2}{35d+10}}{\frac{{d}^{2}-4}{7{d}^{2}-12d-4}}$
$\frac{10{m}^{2}+80m}{3m-9}\xb7\frac{{m}^{2}+4m-21}{{m}^{2}-9m+20}$
$\phantom{\rule{1.5em}{0ex}}\xf7\frac{5{m}^{2}+10m}{2m-10}$
$\frac{4(m+8)(m+7)}{3(m-4)(m+2)}$
$\frac{4{n}^{2}+32n}{3n+2}\xb7\frac{3{n}^{2}-n-2}{{n}^{2}+n-30}$
$\phantom{\rule{1.5em}{0ex}}\xf7\frac{108{n}^{2}-24n}{n+6}$
$\frac{12{p}^{2}+3p}{p+3}\xf7\frac{{p}^{2}+2p-63}{{p}^{2}-p-12}$
$\phantom{\rule{1.5em}{0ex}}\xb7\frac{p-7}{9{p}^{3}-9{p}^{2}}$
$\frac{(4p+1)(p-7)}{3p(p+9)(p-1)}$
$\frac{6q+3}{9{q}^{2}-9q}\xf7\frac{{q}^{2}+14q+33}{{q}^{2}+4q-5}$
$\phantom{\rule{1.5em}{0ex}}\xb7\frac{4{q}^{2}+12q}{12q+6}$
Probability The director of large company is interviewing applicants for two identical jobs. If $w=$ the number of women applicants and $m=$ the number of men applicants, then the probability that two women are selected for the jobs is $\frac{w}{w+m}\xb7\frac{w-1}{w+m-1}.$
ⓐ
$\frac{w\left(w-1\right)}{\left(w+m\right)\left(w+m-1\right)}$
ⓑ
$\frac{2}{21}$
Area of a triangle The area of a triangle with base b and height h is $\frac{bh}{2}.$ If the triangle is stretched to make a new triangle with base and height three times as much as in the original triangle, the area is $\frac{9bh}{2}.$ Calculate how the area of the new triangle compares to the area of the original triangle by dividing $\frac{9bh}{2}$ by $\frac{bh}{2}$ .
Answers will vary.
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?
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