8.2: Multiply and Divide Rational Expressions
- Page ID
- 45076
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We use the same method for multiplying and dividing fractions to multiply and divide rational expressions.
Multiply and Divide Rational Expressions With Monomials
Recall. When we multiply two fractions, we divide out the common factors, e.g.,
\[\dfrac{10}{9}\cdot\dfrac{21}{25}=\dfrac{\cancel{5}\cdot 2}{\cancel{3}\cdot 3}\cdot\dfrac{7\cdot\cancel{3}}{\cancel{5}\cdot 5}=\dfrac{14}{15}\nonumber\]
We multiply rational expressions using the same method.
Multiply: \(\dfrac{25x^2}{9y^8}\cdot\dfrac{24y^4}{55x^7}\)
Solution
Since this is a product of a quotient of monomials, we reduce out common factors and use the rules of exponents.
\[\begin{array}{rl}\dfrac{25x^2}{9y^8}\cdot\dfrac{24y^4}{55x^7}&\text{Multiply across numerators and denominators} \\ \dfrac{25x^2\cdot 24y^4}{9y^8\cdot 55x^7}&\text{Rewrite grouping like-factors} \\ \dfrac{25\cdot 24\cdot x^2\cdot y^4}{9\cdot 55\cdot x^7\cdot y^8}&\text{Reduce out common factors} \\ \dfrac{5\cdot 8}{11\cdot 3\cdot x^5\cdot y^4}&\text{Multiply} \\ \dfrac{40}{33x^5y^4}&\text{Product}\end{array}\nonumber\]
Recall. When we divide two fractions, we change the operation to multiplication and form the reciprocal of the second fraction. Then we multiply the fractions as we did before. E.g.,
\[\dfrac{7}{5}\div\dfrac{14}{15}=\dfrac{7}{5}\cdot\dfrac{15}{14}=\dfrac{\cancel{7}}{\cancel{5}}\cdot\dfrac{3\cdot\cancel{5}}{\cancel{7}\cdot 2}=\dfrac{3}{2}\nonumber\]
We divide rational expressions using the same method.
Divide: \(\dfrac{a^4b^2}{a}\div\dfrac{b^4}{4}\)
Solution
Since this is a quotient of a quotient of monomials, we form the reciprocal of the second fraction and change the division to multiplication, reduce out common factors, and use the rules of exponents.
\[\begin{array}{rl}\dfrac{a^4b^2}{a}\div\dfrac{b^4}{4}&\text{Rewrite the second fraction as its reciprocal} \\ \dfrac{a^4b^2}{a}\cdot\dfrac{4}{b^4}&\text{Multiply across numerators and denominators} \\ \dfrac{4a^4b^2}{ab^4}&\text{Reduce out common factors} \\ \dfrac{4a^3}{b^2}&\text{Quotient} \end{array}\nonumber\]
Multiply and Divide Rational Expressions with Polynomials
When multiplying or dividing polynomials in rational expressions, we first factor using factoring techniques, then reduce out the common factors.
We are not allowed to reduce terms, only factors.
Multiply: \(\dfrac{x^2-9}{x^2+x-20}\cdot\dfrac{x^2-8x+16}{3x+9}\)
Solution
Since we have polynomials in the numerators and denominators, we first factor, then reduce.
\[\begin{array}{rl}\dfrac{x^2-9}{x^2+x-20}\cdot\dfrac{x^2-8x+16}{3x+9}&\text{Factor each numerator and denominator} \\ \dfrac{\color{blue}{(x+3)}\color{black}{}(x-3)}{\color{blue}{(x-4)}\color{black}{}(x+5)}\cdot\dfrac{(x-4)\color{blue}{(x-4)}}{3\color{blue}{(x+3)}}&\color{black}{\text{Reduce out common factors}} \\ \dfrac{\color{blue}{\cancel{(x+3)}}\color{black}{}(x-3)}{\color{blue}{\cancel{(x-4)}}\color{black}{}(x+5)}\cdot\dfrac{(x-4)\color{blue}{\cancel{(x-4)}}}{3\color{blue}{\cancel{(x+3)}}}&\color{black}{\text{Rewrite}} \\ \dfrac{(x-3)}{(x+5)}\cdot\dfrac{(x-4)}{3}&\text{Multiply} \\ \dfrac{(x-3)(x-4)}{3(x+5)}&\text{Product}\end{array}\nonumber\]
We can leave the product in factored form. There’s no reason to multiply out the final answer unless an instructor requests the product that way.
Divide: \(\dfrac{x^2-x-12}{x^2-2x-8}\div\dfrac{5x^2+15x}{x^2+x-2}\)
Solution
Since we have division with polynomials in the numerators and denominators, we form the reciprocal of the second fraction and change the division to multiplication, factor, then reduce out common factors.
\[\begin{array}{rl}\dfrac{x^2-x-12}{x^2-2x-8}\div\dfrac{5x^2+15x}{x^2+x-2}&\text{Rewrite the second fraction as its reciprocal} \\ \dfrac{x^2-x-12}{x^2-2x-8}\cdot\dfrac{x^2+x-2}{5x^2+15x}&\text{Factor each numerator and denominator} \\ \dfrac{\color{blue}{(x-4)(x+3)}}{\color{blue}{(x+2)(x-4)}}\color{black}{}\cdot\dfrac{\color{blue}{(x+2)}\color{black}{}(x-1)}{5x\color{blue}{(x+3)}}&\color{black}{\text{Reduce out common factors}} \\ \dfrac{\color{blue}{\cancel{(x-4)}\cancel{(x+3)}}}{\color{blue}{\cancel{(x+2)}\cancel{(x-4)}}}\color{black}{}\cdot\dfrac{\color{blue}{\cancel{(x+2)}}\color{black}{}(x-1)}{5x\color{blue}{\cancel{(x+3)}}}&\color{black}{\text{Rewrite}} \\ \dfrac{1}{1}\cdot\dfrac{x-1}{5x}&\text{Multiply} \\ \dfrac{(x-1)}{5x}&\text{Quotient}\end{array}\nonumber\]
Multiply and Divide Rational Expressions in General
We can combine multiplying and dividing rational expressions in one expression, but, remember, we form the reciprocal of the fraction that directly proceeds the division sign and then change the division to multiplication. Lastly, we can reduce the common factors.
We are not allowed to reduce terms, only factors.
Simplify: \(\dfrac{a^2+7a+10}{a^2+6a+5}\cdot\dfrac{a+1}{a^2+4a+4}\div\dfrac{a-1}{a+2}\)
Solution
\[\begin{array}{rl}\dfrac{a^2+7a+10}{a^2+6a+5}\cdot\dfrac{a+1}{a^2+4a+4}\div\dfrac{a-1}{a+2}&\text{Form the reciprocal of the last function} \\ \dfrac{a^2+7a+10}{a^2+6a+5}\cdot\dfrac{a+1}{a^2+4a+4}\cdot\dfrac{a+2}{a-1}&\text{Factor each numerator and denominator} \\ \dfrac{\color{blue}{(a+5)(a+2)}}{\color{blue}{(a+5)(a+1)}}\color{black}{}\cdot\dfrac{\color{blue}{(a+1)}}{\color{blue}{(a+2)(a+2)}}\color{black}{}\cdot\dfrac{\color{blue}{(a+2)}}{(a-1)}&\color{black}{\text{Reduce out common factors}} \\ \dfrac{\color{blue}{\cancel{(a+5)}\cancel{(a+2)}}}{\color{blue}{\cancel{(a+5)}\cancel{(a+1)}}}\color{black}{}\cdot\dfrac{\color{blue}{\cancel{(a+1)}}}{\color{blue}{\cancel{(a+2)}\cancel{(a+2)}}}\color{black}{}\cdot\dfrac{\color{blue}{\cancel{(a+2)}}}{(a-1)}&\color{black}{\text{Rewrite}} \\ \dfrac{1}{1}\cdot\dfrac{1}{1}\cdot\dfrac{1}{(a-1)}&\text{Multiply} \\ \dfrac{1}{(a-1)}&\text{Reduced expression}\end{array}\nonumber\]
Indian mathematician Aryabhata, in the \(6^{\text{th}}\) century, published a work which included the rational expression \(\dfrac{n(n + 1)(n + 2)}{6}\) for the sum of the first \(n\) squares \((1^1 + 2^2 + 3^2 +\cdots + n^2 )\)
Multiply and Divide with Rational Functions
Let \(P(x)=\dfrac{4x^2+3x-1}{4x^2+9x+5}\) and \(R(x)=\dfrac{x^2-2x-8}{4x^2+7x-2}\). Find and simplify \((P\cdot R)(x)\).
Solution
First, we apply the definition for the product of two functions, then simplify.
\[\begin{array}{rl}(P\cdot R)(x)=P(x)\cdot R(x)&\text{Replace }P(x)\text{ and }R(x) \\ (P\cdot R)(x)=\dfrac{4x^2+3x-1}{4x^2+9x+5}\cdot\dfrac{x^2-2x-8}{4x^2+7x-2}&\text{Factor each numerator and denominator} \\ (P\cdot R)(x)=\dfrac{(4x-1)(x+1)}{(4x+5)(x+1)}\cdot\dfrac{(x-4)(x+2)}{(4x-1)(x+2)}&\text{Reduce} \\ (P\cdot R)(x)=\dfrac{\cancel{(4x-1)}\cancel{(x+1)}}{(4x+5)\cancel{(x+1)}}\cdot\dfrac{(x-4)\cancel{(x+2)}}{\cancel{(4x-1)}\cancel{(x+2)}}&\text{Rewrite the function} \\ (P\cdot R)(x)=\dfrac{x-4}{4x+5}&\text{Product of }P\text{ and }R\end{array}\nonumber\]
Let \(P(x)=\dfrac{3x^2+14x+8}{3x^2+8x-16}\) and \(R(x)=\dfrac{3x^2-4x-4}{x^2-3x+2}\). Find and simplify \((P\div R)(x)\).
Solution
First, we apply the definition for the division of two functions, then simplify.
\[\begin{array}{rl} (P\div R)(x)=P(x)\div R(x)&\text{Replace }P(x)\text{ and }R(x) \\ (P\div R)(x)=\dfrac{3x^2+14x+8}{3x^2+8x-16}\div\dfrac{3x^2-4x-4}{x^2-3x+2}&\text{Form the reciprocal of }R\text{ and write as multiplication} \\ (P\div R)(x)=\dfrac{3x^2+14x+8}{3x^2+8x-16}\cdot\dfrac{x^2-3x+2}{3x^2-4x-4}&\text{Factor each numerator and denominator} \\ (P\div R)(x)=\dfrac{(3x+2)(x+4)}{(3x-4)(x+4)}\cdot\dfrac{(x-2)(x-1)}{(3x+2)(x-2)}&\text{Reduce} \\ (P\div R)(x)=\dfrac{\cancel{(3x+2)}\cancel{(x+4)}}{(3x-4)\cancel{(x+4)}}\cdot\dfrac{\cancel{(x-2)}(x-1)}{\cancel{(3x+2)}\cancel{(x-2)}}&\text{Rewrite the function} \\ (P\div R)(x)=\dfrac{x-1}{3x-4}&\text{Quotient of }P\text{ and }R\end{array}\nonumber\]
Multiply and Divide Rational Expressions Homework
Simplify each expression.
\(\dfrac{8x^2}{9}\cdot\dfrac{9}{2}\)
\(\dfrac{9n}{2n}\cdot\dfrac{7}{5n}\)
\(\dfrac{5x^2}{4}\cdot\dfrac{6}{5}\)
\(\dfrac{7(m-6)}{m-6}\cdot\dfrac{5m(7m-5)}{7(7m-5)}\)
\(\dfrac{7r}{7r(r+10)}\div\dfrac{r-6}{(r-6)^2}\)
\(\dfrac{25n+25}{5}\cdot\dfrac{4}{30n+30}\)
\(\dfrac{x-10}{35x+21}\div\dfrac{7}{35x+21}\)
\(\dfrac{x^2-6x-7}{x+5}\cdot\dfrac{x+5}{x-7}\)
\(\dfrac{8k}{24k^2-40k}\div\dfrac{1}{15k-25}\)
\((n-8)\cdot\dfrac{6}{10n-80}\)
\(\dfrac{4m+36}{m+9}\cdot\dfrac{m-5}{5m^2}\)
\(\dfrac{3x-6}{12x-24}(x+3)\)
\(\dfrac{b+2}{40b^2-24b}(5b-3)\)
\(\dfrac{n-7}{6n-12}\cdot\dfrac{12-6n}{n^2-13n+42}\)
\(\dfrac{27a+36}{9a+63}\div\dfrac{6a+8}{2}\)
\(\dfrac{x^2-12x+32}{x^2-6x-16}\cdot\dfrac{7x^2+14x}{7x^2+21x}\)
\((10m^2+100m)\cdot\dfrac{18m^3-36m^2}{20m^2-40m}\)
\(\dfrac{7p^2+25p+12}{6p+48}\cdot\dfrac{3p-8}{21p^2-44p-32}\)
\(\dfrac{10b^2}{30b+20}\cdot\dfrac{30b+20}{2b^2+10b}\)
\(\dfrac{7r^2-53r-24}{7r+2}\div\dfrac{49r+21}{49r+14}\)
\(\dfrac{8x}{3x}\div\dfrac{4}{7}\)
\(\dfrac{9m}{5m^2}\cdot\dfrac{7}{2}\)
\(\dfrac{10p}{5}\div\dfrac{8}{10}\)
\(\dfrac{7}{10(n+3)}\div\dfrac{n-2}{(n+3)(n-2)}\)
\(\dfrac{6x(x+4)}{x-3}\cdot\dfrac{(x-3)(x-6)}{6x(x-6)}\)
\(\dfrac{9}{b^2-b-12}\div\dfrac{b-5}{b^2-b-12}\)
\(\dfrac{v-1}{4}\cdot\dfrac{4}{v^2-11v+10}\)
\(\dfrac{1}{a-6}\cdot\dfrac{8a+80}{8}\)
\(\dfrac{p-8}{p^2-12p+32}\div\dfrac{1}{p-10}\)
\(\dfrac{x^2-7x+10}{x-2}\cdot\dfrac{x+10}{x^2-x-20}\)
\(\dfrac{2r}{r+6}\div\dfrac{2r}{7r+42}\)
\(\dfrac{2n^2-12n-54}{n+7}\div (2n+6)\)
\(\dfrac{21v^2+16v-16}{3v+4}\div\dfrac{35v-20}{v-9}\)
\(\dfrac{x^2+11x+24}{6x^3+18x^2}\cdot\dfrac{6x^3+6x^2}{x^2+5x-24}\)
\(\dfrac{k-7}{k^2-k-12}\cdot\dfrac{7k^2-28k}{8k^2-56k}\)
\(\dfrac{9x^3+54x^2}{x^2+5x-14}\cdot\dfrac{x^2+5x-14}{10x^2}\)
\(\dfrac{n-7}{n^2-2n-35}\div\dfrac{9n+54}{10n+50}\)
\(\dfrac{7x^2-66x+80}{49x^2+7x-72}\div\dfrac{7x^2+39x-70}{49x^2+7x-72}\)
\(\dfrac{35n^2-12n-32}{49n^2-91n+40}\cdot\dfrac{7n^2+16n-15}{5n+4}\)
\(\dfrac{12x+24}{10x^2+34x+28}\cdot\dfrac{15x+21}{5}\)
\(\dfrac{x^2-1}{2x-4}\cdot\dfrac{x^2-4}{x^2-x-2}\div\dfrac{x^2+x-2}{3x-6}\)
\(\dfrac{x^2+3x+9}{x^2+x-12}\cdot\dfrac{x^2+2x-8}{x^3-27}\div\dfrac{x^2-4}{x^2-6x+9}\)
\(\dfrac{a^3+b^3}{a^2+3ab+2b^2}\cdot\dfrac{3a-6b}{3a^2-3ab+3b^2}\div\dfrac{a^2-4b^2}{a+2b}\)
\(\dfrac{x^2+3x-10}{x^2+6x+5}\cdot\dfrac{2x^2-x-3}{2x^2+x-6}\div\dfrac{8x+20}{6x+15}\)
Perform the indicated operation and simplify.
Let \(f(x)=\dfrac{5x^2+8x+3}{5x^2+7x+2}\) and \(g(x)=\dfrac{x^2-4x+3}{5x^2-2x-3}\). Find and simplify \((P\cdot R)(x)\).
Let \(f(x)=\dfrac{4x^2-21x+5}{4x^2-23x+15}\) and \(g(x)=\dfrac{x^2+5x+6}{4x^2+11x-3}\). Find and simplify \((P\cdot R)(x)\).
Let \(P(x)=\dfrac{3x^2-10x+8}{3x^2-4x-4}\) and \(R(x)=\dfrac{3x^2+8x-16}{x^2+5x+4}\). Find and simplify \((P\div R)(x)\).
Let \(P(x)=\dfrac{4x^2+19x-5}{4x^2+17x-15}\) and \(R(x)=\dfrac{4x^2-21x+5}{x^2-3x-10}\). Find and simplify \((P\div R)(x)\).