# 8.4 Building rational expressions and the lcd  (Page 4/4)

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$\begin{array}{lll}\frac{4b}{b-1},\frac{-2b}{b+3}.\hfill & \hfill & \text{By\hspace{0.17em}inspection,\hspace{0.17em}the\hspace{0.17em}LCD\hspace{0.17em}is\hspace{0.17em}}\left(b-1\right)\left(b+3\right).\hfill \\ \hfill & \hfill & \text{Rewrite\hspace{0.17em}each\hspace{0.17em}fraction\hspace{0.17em}with\hspace{0.17em}new\hspace{0.17em}denominator\hspace{0.17em}}\left(b-1\right)\left(b+3\right).\hfill \\ \frac{}{\left(b-1\right)\left(b+3\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{}{\left(b-1\right)\left(b+3\right)}\hfill & \hfill & \begin{array}{l}\text{The\hspace{0.17em}denominator\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}first\hspace{0.17em}rational\hspace{0.17em}expression\hspace{0.17em}has\hspace{0.17em}been\hspace{0.17em}multiplied\hspace{0.17em}}\\ \text{by\hspace{0.17em}}b\text{\hspace{0.17em}}+3,\text{\hspace{0.17em}}\text{so\hspace{0.17em}the\hspace{0.17em}numerator\hspace{0.17em}}4b\text{\hspace{0.17em}must\hspace{0.17em}be\hspace{0.17em}multiplied\hspace{0.17em}by\hspace{0.17em}}b\text{\hspace{0.17em}}+3.\text{\hspace{0.17em}}\end{array}\hfill \\ \hfill & \hfill & 4b\left(b+3\right)=4{b}^{2}+12b\hfill \\ \frac{4{b}^{2}+12b}{\left(b-1\right)\left(b+3\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{}{\left(b-1\right)\left(b+3\right)}\hfill & \hfill & \begin{array}{l}\text{The\hspace{0.17em}denominator\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}second\hspace{0.17em}rational\hspace{0.17em}expression\hspace{0.17em}has\hspace{0.17em}been\hspace{0.17em}multiplied\hspace{0.17em}}\\ \text{by\hspace{0.17em}}b-1\text{,\hspace{0.17em}so\hspace{0.17em}the\hspace{0.17em}numerator\hspace{0.17em}}-2b\text{\hspace{0.17em}must\hspace{0.17em}be\hspace{0.17em}multiplied\hspace{0.17em}by\hspace{0.17em}}b-1.\end{array}\hfill \\ \hfill & \hfill & -2b\left(b-1\right)=-2{b}^{2}+2b\hfill \\ \frac{4{b}^{2}+12b}{\left(b-1\right)\left(b+3\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{-2{b}^{2}+2b}{\left(b-1\right)\left(b+3\right)}\hfill & \hfill & \hfill \end{array}$

$\begin{array}{lll}\frac{6x}{{x}^{2}-8x+15},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{-2{x}^{2}}{{x}^{2}-7x+12}.\hfill & \hfill & \text{We\hspace{0.17em}first\hspace{0.17em}find\hspace{0.17em}the\hspace{0.17em}LCD}.\text{\hspace{0.17em}Factor}.\text{\hspace{0.17em}}\hfill \\ \frac{6x}{\left(x-3\right)\left(x-5\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{-2{x}^{2}}{\left(x-3\right)\left(x-4\right)}\hfill & \hfill & \begin{array}{l}\text{The\hspace{0.17em}LCD\hspace{0.17em}is\hspace{0.17em}}\left(x-3\right)\left(x-5\right)\left(x-4\right).\text{\hspace{0.17em}}\text{Rewrite\hspace{0.17em}each\hspace{0.17em}of\hspace{0.17em}these\hspace{0.17em}}\\ \text{fractions\hspace{0.17em}with\hspace{0.17em}new\hspace{0.17em}denominator\hspace{0.17em}}\left(x-3\right)\left(x-5\right)\left(x-4\right).\end{array}\hfill \\ \frac{}{\left(x-3\right)\left(x-5\right)\left(x-4\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{}{\left(x-3\right)\left(x-5\right)\left(x-4\right)}\hfill & \hfill & \begin{array}{l}\text{By\hspace{0.17em}comparing\hspace{0.17em}the\hspace{0.17em}denominator\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}first\hspace{0.17em}fraction\hspace{0.17em}with\hspace{0.17em}the\hspace{0.17em}LCD\hspace{0.17em}}\\ \text{we\hspace{0.17em}see\hspace{0.17em}that\hspace{0.17em}we\hspace{0.17em}must\hspace{0.17em}multiply\hspace{0.17em}the\hspace{0.17em}numerator\hspace{0.17em}}6x\text{\hspace{0.17em}by\hspace{0.17em}}x-4.\end{array}\hfill \\ \hfill & \hfill & 6x\left(x-4\right)=6{x}^{2}-24x\hfill \\ \frac{6{x}^{2}-24x}{\left(x-3\right)\left(x-5\right)\left(x-4\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{}{\left(x-3\right)\left(x-5\right)\left(x-4\right)}\hfill & \hfill & \begin{array}{l}\text{By\hspace{0.17em}comparing\hspace{0.17em}the\hspace{0.17em}denominator\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}second\hspace{0.17em}fraction\hspace{0.17em}with\hspace{0.17em}the\hspace{0.17em}LCD,\hspace{0.17em}}\\ \text{we\hspace{0.17em}see\hspace{0.17em}that\hspace{0.17em}we\hspace{0.17em}must\hspace{0.17em}multiply\hspace{0.17em}the\hspace{0.17em}numerator\hspace{0.17em}}-2{x}^{2}\text{\hspace{0.17em}by\hspace{0.17em}}x-5.\end{array}\hfill \\ \hfill & \hfill & -2{x}^{2}\left(x-5\right)=-2{x}^{3}+10{x}^{2}\hfill \\ \hfill & \hfill & \hfill \\ \frac{6{x}^{2}-24x}{\left(x-3\right)\left(x-5\right)\left(x-4\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{-2{x}^{3}+10{x}^{2}}{\left(x-3\right)\left(x-5\right)\left(x-4\right)}\hfill & \hfill & \hfill \end{array}$

These examples have been done step-by-step and include explanations. This makes the process seem fairly long. In practice, however, the process is much quicker.

$\begin{array}{lll}\frac{6ab}{{a}^{2}-5a+4},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{a+b}{{a}^{2}-8a+16}\hfill & \hfill & \hfill \\ \frac{6ab}{\left(a-1\right)\left(a-4\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{a+b}{{\left(a-4\right)}^{2}}\hfill & \hfill & \text{LCD}\text{\hspace{0.17em}}=\left(a-1\right){\left(a-4\right)}^{2}.\hfill \\ \frac{6ab\left(a-4\right)}{\left(a-1\right){\left(a-4\right)}^{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\left(a+b\right)\left(a-1\right)}{\left(a-1\right){\left(a-4\right)}^{2}}\hfill & \hfill & \hfill \end{array}$

$\begin{array}{l}\begin{array}{lll}\frac{x+1}{{x}^{3}+3{x}^{2}},\frac{2x}{{x}^{3}-4x},\frac{x-4}{{x}^{2}-4x+4}\hfill & \hfill & \hfill \\ \frac{x+1}{{x}^{2}\left(x+3\right)},\frac{2x}{x\left(x+2\right)\left(x-2\right)},\frac{x-4}{{\left(x-2\right)}^{2}}\hfill & \hfill & \text{LCD}={x}^{2}\left(x+3\right)\left(x+2\right){\left(x-2\right)}^{2}.\hfill \end{array}\\ \frac{\left(x+1\right)\left(x+2\right){\left(x-2\right)}^{2}}{{x}^{2}\left(x+3\right)\left(x+2\right){\left(x-2\right)}^{2}},\frac{2{x}^{2}\left(x+3\right)\left(x-2\right)}{{x}^{2}\left(x+3\right)\left(x+2\right){\left(x-2\right)}^{2}},\frac{{x}^{2}\left(x+3\right)\left(x+2\right)\left(x-4\right)}{{x}^{2}\left(x+3\right)\left(x+2\right){\left(x-2\right)}^{2}}\end{array}$

## Practice set c

Change the given rational expressions into rational expressions with the same denominators.

$\frac{4}{{x}^{3}},\frac{7}{{x}^{5}}$

$\frac{4{x}^{2}}{{x}^{5}},\frac{7}{{x}^{5}}$

$\frac{2x}{x+6},\frac{x}{x-1}$

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

$\frac{-3}{{b}^{2}-b},\frac{4b}{{b}^{2}-1}$

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

$\frac{8}{{x}^{2}-x-6},\frac{-1}{{x}^{2}+x-2}$

$\frac{8\left(x-1\right)}{\left(x-3\right)\left(x+2\right)\left(x-1\right)},\frac{-1\left(x-3\right)}{\left(x-3\right)\left(x+2\right)\left(x-1\right)}$

$\frac{10x}{{x}^{2}+8x+16},\frac{5x}{{x}^{2}-16}$

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

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

$\frac{-2{a}^{2}{b}^{2}{\left(a-2\right)}^{2}}{{a}^{3}\left(a-6\right){\left(a-2\right)}^{2}},\frac{6b\left(a-6\right)\left(a-2\right)}{{a}^{3}\left(a-6\right){\left(a-2\right)}^{2}},\frac{-2{a}^{4}\left(a-6\right)}{{a}^{3}\left(a-6\right){\left(a-2\right)}^{2}}$

## Exercises

For the following problems, replace $N$ with the proper quantity.

$\frac{3}{x}=\frac{N}{{x}^{3}}$

$3{x}^{2}$

$\frac{4}{a}=\frac{N}{{a}^{2}}$

$\frac{-2}{x}=\frac{N}{xy}$

$-2y$

$\frac{-7}{m}=\frac{N}{ms}$

$\frac{6a}{5}=\frac{N}{10b}$

$12ab$

$\frac{a}{3z}=\frac{N}{12z}$

$\frac{{x}^{2}}{4{y}^{2}}=\frac{N}{20{y}^{4}}$

$5{x}^{2}{y}^{2}$

$\frac{{b}^{3}}{6a}=\frac{N}{18{a}^{5}}$

$\frac{-4a}{5{x}^{2}y}=\frac{N}{15{x}^{3}{y}^{3}}$

$-12ax{y}^{2}$

$\frac{-10z}{7{a}^{3}b}=\frac{N}{21{a}^{4}{b}^{5}}$

$\frac{8{x}^{2}y}{5{a}^{3}}=\frac{N}{25{a}^{3}{x}^{2}}$

$40{x}^{4}y$

$\frac{2}{{a}^{2}}=\frac{N}{{a}^{2}\left(a-1\right)}$

$\frac{5}{{x}^{3}}=\frac{N}{{x}^{3}\left(x-2\right)}$

$5\left(x-2\right)$

$\frac{2a}{{b}^{2}}=\frac{N}{{b}^{3}-b}$

$\frac{4x}{a}=\frac{N}{{a}^{4}-4{a}^{2}}$

$4ax\left(a+2\right)\left(a-2\right)$

$\frac{6{b}^{3}}{5a}=\frac{N}{10{a}^{2}-30a}$

$\frac{4x}{3b}=\frac{N}{3{b}^{5}-15b}$

$4x\left({b}^{4}-5\right)$

$\frac{2m}{m-1}=\frac{N}{\left(m-1\right)\left(m+2\right)}$

$\frac{3s}{s+12}=\frac{N}{\left(s+12\right)\left(s-7\right)}$

$3s\left(s-7\right)$

$\frac{a+1}{a-3}=\frac{N}{\left(a-3\right)\left(a-4\right)}$

$\frac{a+2}{a-2}=\frac{N}{\left(a-2\right)\left(a-4\right)}$

$\left(a+2\right)\left(a-4\right)$

$\frac{b+7}{b-6}=\frac{N}{\left(b-6\right)\left(b+6\right)}$

$\frac{5m}{2m+1}=\frac{N}{\left(2m+1\right)\left(m-2\right)}$

$5m\left(m-2\right)$

$\frac{4}{a+6}=\frac{N}{{a}^{2}+5a-6}$

$\frac{9}{b-2}=\frac{N}{{b}^{2}-6b+8}$

$9\left(b-4\right)$

$\frac{3b}{b-3}=\frac{N}{{b}^{2}-11b+24}$

$\frac{-2x}{x-7}=\frac{N}{{x}^{2}-4x-21}$

$-2x\left(x+3\right)$

$\frac{-6m}{m+6}=\frac{N}{{m}^{2}+10m+24}$

$\frac{4y}{y+1}=\frac{N}{{y}^{2}+9y+8}$

$4y\left(y+8\right)$

$\frac{x+2}{x-2}=\frac{N}{{x}^{2}-4}$

$\frac{y-3}{y+3}=\frac{N}{{y}^{2}-9}$

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

$\frac{a+5}{a-5}=\frac{N}{{a}^{2}-25}$

$\frac{z-4}{z+4}=\frac{N}{{z}^{2}-16}$

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

$\frac{4}{2a+1}=\frac{N}{2{a}^{2}-5a-3}$

$\frac{1}{3b-1}=\frac{N}{3{b}^{2}+11b-4}$

$b+4$

$\frac{a+2}{2a-1}=\frac{N}{2{a}^{2}+9a-5}$

$\frac{-3}{4x+3}=\frac{N}{4{x}^{2}-13x-12}$

$-3\left(x-4\right)$

$\frac{b+2}{3b-1}=\frac{N}{6{b}^{2}+7b-3}$

$\frac{x-1}{4x-5}=\frac{N}{12{x}^{2}-11x-5}$

$\left(x-1\right)\left(3x+1\right)$

$\frac{3}{x+2}=\frac{3x-21}{N}$

$\frac{4}{y+6}=\frac{4y+8}{N}$

$\left(y+6\right)\left(y+2\right)$

$\frac{-6}{a-1}=\frac{-6a-18}{N}$

$\frac{-8a}{a+3}=\frac{-8{a}^{2}-40a}{N}$

$\left(a+3\right)\left(a+5\right)$

$\frac{y+1}{y-8}=\frac{{y}^{2}-2y-3}{N}$

$\frac{x-4}{x+9}=\frac{{x}^{2}+x-20}{N}$

$\left(x+9\right)\left(x+5\right)$

$\frac{3x}{2-x}=\frac{N}{x-2}$

$\frac{7a}{5-a}=\frac{N}{a-5}$

$-7a$

$\frac{-m+1}{3-m}=\frac{N}{m-3}$

$\frac{k+6}{10-k}=\frac{N}{k-10}$

$-k-6$

For the following problems, convert the given rational expressions to rational expressions having the same denominators.

$\frac{2}{a},\frac{3}{{a}^{4}}$

$\frac{5}{{b}^{2}},\frac{4}{{b}^{3}}$

$\frac{5b}{{b}^{3}},\frac{4}{{b}^{3}}$

$\frac{8}{z},\frac{3}{4{z}^{3}}$

$\frac{9}{{x}^{2}},\frac{1}{4x}$

$\frac{36}{4{x}^{2}},\frac{x}{4{x}^{2}}$

$\frac{2}{a+3},\frac{4}{a+1}$

$\frac{2}{x+5},\frac{4}{x-5}$

$\frac{2\left(x-5\right)}{\left(x+5\right)\left(x-5\right)},\frac{4\left(x+5\right)}{\left(x+5\right)\left(x-5\right)}$

$\frac{1}{x-7},\frac{4}{x-1}$

$\frac{10}{y+2},\frac{1}{y+8}$

$\frac{10\left(y+8\right)}{\left(y+2\right)\left(y+8\right)},\frac{y+2}{\left(y+2\right)\left(y+8\right)}$

$\frac{4}{{a}^{2}},\frac{a}{a+4}$

$\frac{-3}{{b}^{2}},\frac{{b}^{2}}{b+5}$

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

$\frac{-6}{b-1},\frac{5b}{4b}$

$\frac{10a}{a-6},\frac{2}{{a}^{2}-6a}$

$\frac{10{a}^{2}}{a\left(a-6\right)},\frac{2}{a\left(a-6\right)}$

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

$\frac{x+1}{{x}^{2}-x-6},\frac{x+4}{{x}^{2}+x-2}$

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

$\frac{x-5}{{x}^{2}-9x+20},\frac{4}{{x}^{2}-3x-10}$

$\frac{-4}{{b}^{2}+5b-6},\frac{b+6}{{b}^{2}-1}$

$\frac{-4\left(b+1\right)}{\left(b+1\right)\left(b-1\right)\left(b+6\right)},\frac{{\left(b+6\right)}^{2}}{\left(b+1\right)\left(b-1\right)\left(b+6\right)}$

$\frac{b+2}{{b}^{2}+6b+8},\frac{b-1}{{b}^{2}+8b+12}$

$\frac{x+7}{{x}^{2}-2x-3},\frac{x+3}{{x}^{2}-6x-7}$

$\frac{\left(x+7\right)\left(x-7\right)}{\left(x+1\right)\left(x-3\right)\left(x-7\right)},\frac{\left(x+3\right)\left(x-3\right)}{\left(x+1\right)\left(x-3\right)\left(x-7\right)}$

$\frac{2}{{a}^{2}+a},\frac{a+3}{{a}^{2}-1}$

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

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

$\frac{x-2}{2{x}^{2}+5x-3},\frac{x.-1}{5{x}^{2}+16x+3}$

$\frac{2}{x-5},\frac{-3}{5-x}$

$\frac{2}{x-5},\frac{3}{x-5}$

$\frac{4}{a-6},\frac{-5}{6-a}$

$\frac{6}{2-x},\frac{5}{x-2}$

$\frac{-6}{x-2},\frac{5}{x-2}$

$\frac{k}{5-k},\frac{3k}{k-5}$

$\frac{2m}{m-8},\frac{7}{8-m}$

$\frac{2m}{m-8},\frac{-7}{m-8}$

## Excercises for review

( [link] ) Factor ${m}^{2}{x}^{3}+m{x}^{2}+mx.$

( [link] ) Factor ${y}^{2}-10y+21.$

$\left(y-7\right)\left(y-3\right)$

( [link] ) Write the equation of the line that passes through the points $\left(1,\text{\hspace{0.17em}}1\right)$ and $\left(4,\text{\hspace{0.17em}}-2\right)$ . Express the equation in slope-intercept form.

( [link] ) Reduce $\frac{{y}^{2}-y-6}{y-3}.$

$y+2$

( [link] ) Find the quotient: $\frac{{x}^{2}-6x+9}{{x}^{2}-x-6}÷\frac{{x}^{2}+2x-15}{{x}^{2}+2x}.$

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