8.2: Multiplicar y dividir expresiones racionales
- Page ID
- 117375
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Utilizamos el mismo método para multiplicar y dividir fracciones para multiplicar y dividir expresiones racionales.
Multiplicar y dividir expresiones racionales con monomios
Recordar. Cuando multiplicamos dos fracciones, dividimos los factores comunes, e.g.
\[\dfrac{10}{9}\cdot\dfrac{21}{25}=\dfrac{\cancel{5}\cdot 2}{\cancel{3}\cdot 3}\cdot\dfrac{7\cdot\cancel{3}}{\cancel{5}\cdot 5}=\dfrac{14}{15}\nonumber\]
Multiplicamos expresiones racionales usando el mismo método.
Multiplicar:\(\dfrac{25x^2}{9y^8}\cdot\dfrac{24y^4}{55x^7}\)
Solución
Dado que esto es producto de un cociente de monomios, reducimos factores comunes y utilizamos las reglas de exponentes.
\[\begin{array}{rl}\dfrac{25x^2}{9y^8}\cdot\dfrac{24y^4}{55x^7}&\text{Multiply across numerators and denominators} \\ \dfrac{25x^2\cdot 24y^4}{9y^8\cdot 55x^7}&\text{Rewrite grouping like-factors} \\ \dfrac{25\cdot 24\cdot x^2\cdot y^4}{9\cdot 55\cdot x^7\cdot y^8}&\text{Reduce out common factors} \\ \dfrac{5\cdot 8}{11\cdot 3\cdot x^5\cdot y^4}&\text{Multiply} \\ \dfrac{40}{33x^5y^4}&\text{Product}\end{array}\nonumber\]
Recordar. Cuando dividimos dos fracciones, cambiamos la operación a multiplicación y formamos el recíproco de la segunda fracción. Entonces multiplicamos las fracciones como lo hacíamos antes. Por Ejemplo,
\[\dfrac{7}{5}\div\dfrac{14}{15}=\dfrac{7}{5}\cdot\dfrac{15}{14}=\dfrac{\cancel{7}}{\cancel{5}}\cdot\dfrac{3\cdot\cancel{5}}{\cancel{7}\cdot 2}=\dfrac{3}{2}\nonumber\]
Dividimos expresiones racionales usando el mismo método.
Dividir:\(\dfrac{a^4b^2}{a}\div\dfrac{b^4}{4}\)
Solución
Dado que este es un cociente de un cociente de monomios, formamos el recíproco de la segunda fracción y cambiamos la división a multiplicación, reducimos factores comunes y usamos las reglas de exponentes.
\[\begin{array}{rl}\dfrac{a^4b^2}{a}\div\dfrac{b^4}{4}&\text{Rewrite the second fraction as its reciprocal} \\ \dfrac{a^4b^2}{a}\cdot\dfrac{4}{b^4}&\text{Multiply across numerators and denominators} \\ \dfrac{4a^4b^2}{ab^4}&\text{Reduce out common factors} \\ \dfrac{4a^3}{b^2}&\text{Quotient} \end{array}\nonumber\]
Multiplicar y dividir expresiones racionales con polinomios
Al multiplicar o dividir polinomios en expresiones racionales, primero factorizamos usando técnicas de factorización, luego reducimos los factores comunes.
No se nos permite reducir términos, solo factores.
Multiplicar:\(\dfrac{x^2-9}{x^2+x-20}\cdot\dfrac{x^2-8x+16}{3x+9}\)
Solución
Ya que tenemos polinomios en los numeradores y denominadores, primero factorizamos, luego reducimos.
\[\begin{array}{rl}\dfrac{x^2-9}{x^2+x-20}\cdot\dfrac{x^2-8x+16}{3x+9}&\text{Factor each numerator and denominator} \\ \dfrac{\color{blue}{(x+3)}\color{black}{}(x-3)}{\color{blue}{(x-4)}\color{black}{}(x+5)}\cdot\dfrac{(x-4)\color{blue}{(x-4)}}{3\color{blue}{(x+3)}}&\color{black}{\text{Reduce out common factors}} \\ \dfrac{\color{blue}{\cancel{(x+3)}}\color{black}{}(x-3)}{\color{blue}{\cancel{(x-4)}}\color{black}{}(x+5)}\cdot\dfrac{(x-4)\color{blue}{\cancel{(x-4)}}}{3\color{blue}{\cancel{(x+3)}}}&\color{black}{\text{Rewrite}} \\ \dfrac{(x-3)}{(x+5)}\cdot\dfrac{(x-4)}{3}&\text{Multiply} \\ \dfrac{(x-3)(x-4)}{3(x+5)}&\text{Product}\end{array}\nonumber\]
Podemos dejar el producto en forma factorizada. No hay razón para multiplicar la respuesta final a menos que un instructor solicite el producto de esa manera.
Dividir:\(\dfrac{x^2-x-12}{x^2-2x-8}\div\dfrac{5x^2+15x}{x^2+x-2}\)
Solución
Dado que tenemos división con polinomios en los numeradores y denominadores, formamos el recíproco de la segunda fracción y cambiamos la división a multiplicación, factor, luego reducimos factores comunes.
\[\begin{array}{rl}\dfrac{x^2-x-12}{x^2-2x-8}\div\dfrac{5x^2+15x}{x^2+x-2}&\text{Rewrite the second fraction as its reciprocal} \\ \dfrac{x^2-x-12}{x^2-2x-8}\cdot\dfrac{x^2+x-2}{5x^2+15x}&\text{Factor each numerator and denominator} \\ \dfrac{\color{blue}{(x-4)(x+3)}}{\color{blue}{(x+2)(x-4)}}\color{black}{}\cdot\dfrac{\color{blue}{(x+2)}\color{black}{}(x-1)}{5x\color{blue}{(x+3)}}&\color{black}{\text{Reduce out common factors}} \\ \dfrac{\color{blue}{\cancel{(x-4)}\cancel{(x+3)}}}{\color{blue}{\cancel{(x+2)}\cancel{(x-4)}}}\color{black}{}\cdot\dfrac{\color{blue}{\cancel{(x+2)}}\color{black}{}(x-1)}{5x\color{blue}{\cancel{(x+3)}}}&\color{black}{\text{Rewrite}} \\ \dfrac{1}{1}\cdot\dfrac{x-1}{5x}&\text{Multiply} \\ \dfrac{(x-1)}{5x}&\text{Quotient}\end{array}\nonumber\]
Multiplicar y dividir expresiones racionales en general
Podemos combinar multiplicar y dividir expresiones racionales en una sola expresión, pero, recuerden, formamos el recíproco de la fracción que procede directamente del signo de división y luego cambiamos la división a multiplicación. Por último, podemos reducir los factores comunes.
No se nos permite reducir términos, solo factores.
Simplificar:\(\dfrac{a^2+7a+10}{a^2+6a+5}\cdot\dfrac{a+1}{a^2+4a+4}\div\dfrac{a-1}{a+2}\)
Solución
\[\begin{array}{rl}\dfrac{a^2+7a+10}{a^2+6a+5}\cdot\dfrac{a+1}{a^2+4a+4}\div\dfrac{a-1}{a+2}&\text{Form the reciprocal of the last function} \\ \dfrac{a^2+7a+10}{a^2+6a+5}\cdot\dfrac{a+1}{a^2+4a+4}\cdot\dfrac{a+2}{a-1}&\text{Factor each numerator and denominator} \\ \dfrac{\color{blue}{(a+5)(a+2)}}{\color{blue}{(a+5)(a+1)}}\color{black}{}\cdot\dfrac{\color{blue}{(a+1)}}{\color{blue}{(a+2)(a+2)}}\color{black}{}\cdot\dfrac{\color{blue}{(a+2)}}{(a-1)}&\color{black}{\text{Reduce out common factors}} \\ \dfrac{\color{blue}{\cancel{(a+5)}\cancel{(a+2)}}}{\color{blue}{\cancel{(a+5)}\cancel{(a+1)}}}\color{black}{}\cdot\dfrac{\color{blue}{\cancel{(a+1)}}}{\color{blue}{\cancel{(a+2)}\cancel{(a+2)}}}\color{black}{}\cdot\dfrac{\color{blue}{\cancel{(a+2)}}}{(a-1)}&\color{black}{\text{Rewrite}} \\ \dfrac{1}{1}\cdot\dfrac{1}{1}\cdot\dfrac{1}{(a-1)}&\text{Multiply} \\ \dfrac{1}{(a-1)}&\text{Reduced expression}\end{array}\nonumber\]
El matemático indio Aryabhata, en el\(6^{\text{th}}\) siglo, publicó una obra que incluía la expresión racional\(\dfrac{n(n + 1)(n + 2)}{6}\) para la suma de los primeros\(n\) cuadrados\((1^1 + 2^2 + 3^2 +\cdots + n^2 )\)
Multiplicar y dividir con funciones racionales
Dejar\(P(x)=\dfrac{4x^2+3x-1}{4x^2+9x+5}\) y\(R(x)=\dfrac{x^2-2x-8}{4x^2+7x-2}\). Encuentra y simplifica\((P\cdot R)(x)\).
Solución
Primero, aplicamos la definición para el producto de dos funciones, luego simplificamos.
\[\begin{array}{rl}(P\cdot R)(x)=P(x)\cdot R(x)&\text{Replace }P(x)\text{ and }R(x) \\ (P\cdot R)(x)=\dfrac{4x^2+3x-1}{4x^2+9x+5}\cdot\dfrac{x^2-2x-8}{4x^2+7x-2}&\text{Factor each numerator and denominator} \\ (P\cdot R)(x)=\dfrac{(4x-1)(x+1)}{(4x+5)(x+1)}\cdot\dfrac{(x-4)(x+2)}{(4x-1)(x+2)}&\text{Reduce} \\ (P\cdot R)(x)=\dfrac{\cancel{(4x-1)}\cancel{(x+1)}}{(4x+5)\cancel{(x+1)}}\cdot\dfrac{(x-4)\cancel{(x+2)}}{\cancel{(4x-1)}\cancel{(x+2)}}&\text{Rewrite the function} \\ (P\cdot R)(x)=\dfrac{x-4}{4x+5}&\text{Product of }P\text{ and }R\end{array}\nonumber\]
Dejar\(P(x)=\dfrac{3x^2+14x+8}{3x^2+8x-16}\) y\(R(x)=\dfrac{3x^2-4x-4}{x^2-3x+2}\). Encuentra y simplifica\((P\div R)(x)\).
Solución
Primero, aplicamos la definición para la división de dos funciones, luego simplificamos.
\[\begin{array}{rl} (P\div R)(x)=P(x)\div R(x)&\text{Replace }P(x)\text{ and }R(x) \\ (P\div R)(x)=\dfrac{3x^2+14x+8}{3x^2+8x-16}\div\dfrac{3x^2-4x-4}{x^2-3x+2}&\text{Form the reciprocal of }R\text{ and write as multiplication} \\ (P\div R)(x)=\dfrac{3x^2+14x+8}{3x^2+8x-16}\cdot\dfrac{x^2-3x+2}{3x^2-4x-4}&\text{Factor each numerator and denominator} \\ (P\div R)(x)=\dfrac{(3x+2)(x+4)}{(3x-4)(x+4)}\cdot\dfrac{(x-2)(x-1)}{(3x+2)(x-2)}&\text{Reduce} \\ (P\div R)(x)=\dfrac{\cancel{(3x+2)}\cancel{(x+4)}}{(3x-4)\cancel{(x+4)}}\cdot\dfrac{\cancel{(x-2)}(x-1)}{\cancel{(3x+2)}\cancel{(x-2)}}&\text{Rewrite the function} \\ (P\div R)(x)=\dfrac{x-1}{3x-4}&\text{Quotient of }P\text{ and }R\end{array}\nonumber\]
Multiplicar y dividir las expresiones racionales
Simplifica cada expresión.
\(\dfrac{8x^2}{9}\cdot\dfrac{9}{2}\)
\(\dfrac{9n}{2n}\cdot\dfrac{7}{5n}\)
\(\dfrac{5x^2}{4}\cdot\dfrac{6}{5}\)
\(\dfrac{7(m-6)}{m-6}\cdot\dfrac{5m(7m-5)}{7(7m-5)}\)
\(\dfrac{7r}{7r(r+10)}\div\dfrac{r-6}{(r-6)^2}\)
\(\dfrac{25n+25}{5}\cdot\dfrac{4}{30n+30}\)
\(\dfrac{x-10}{35x+21}\div\dfrac{7}{35x+21}\)
\(\dfrac{x^2-6x-7}{x+5}\cdot\dfrac{x+5}{x-7}\)
\(\dfrac{8k}{24k^2-40k}\div\dfrac{1}{15k-25}\)
\((n-8)\cdot\dfrac{6}{10n-80}\)
\(\dfrac{4m+36}{m+9}\cdot\dfrac{m-5}{5m^2}\)
\(\dfrac{3x-6}{12x-24}(x+3)\)
\(\dfrac{b+2}{40b^2-24b}(5b-3)\)
\(\dfrac{n-7}{6n-12}\cdot\dfrac{12-6n}{n^2-13n+42}\)
\(\dfrac{27a+36}{9a+63}\div\dfrac{6a+8}{2}\)
\(\dfrac{x^2-12x+32}{x^2-6x-16}\cdot\dfrac{7x^2+14x}{7x^2+21x}\)
\((10m^2+100m)\cdot\dfrac{18m^3-36m^2}{20m^2-40m}\)
\(\dfrac{7p^2+25p+12}{6p+48}\cdot\dfrac{3p-8}{21p^2-44p-32}\)
\(\dfrac{10b^2}{30b+20}\cdot\dfrac{30b+20}{2b^2+10b}\)
\(\dfrac{7r^2-53r-24}{7r+2}\div\dfrac{49r+21}{49r+14}\)
\(\dfrac{8x}{3x}\div\dfrac{4}{7}\)
\(\dfrac{9m}{5m^2}\cdot\dfrac{7}{2}\)
\(\dfrac{10p}{5}\div\dfrac{8}{10}\)
\(\dfrac{7}{10(n+3)}\div\dfrac{n-2}{(n+3)(n-2)}\)
\(\dfrac{6x(x+4)}{x-3}\cdot\dfrac{(x-3)(x-6)}{6x(x-6)}\)
\(\dfrac{9}{b^2-b-12}\div\dfrac{b-5}{b^2-b-12}\)
\(\dfrac{v-1}{4}\cdot\dfrac{4}{v^2-11v+10}\)
\(\dfrac{1}{a-6}\cdot\dfrac{8a+80}{8}\)
\(\dfrac{p-8}{p^2-12p+32}\div\dfrac{1}{p-10}\)
\(\dfrac{x^2-7x+10}{x-2}\cdot\dfrac{x+10}{x^2-x-20}\)
\(\dfrac{2r}{r+6}\div\dfrac{2r}{7r+42}\)
\(\dfrac{2n^2-12n-54}{n+7}\div (2n+6)\)
\(\dfrac{21v^2+16v-16}{3v+4}\div\dfrac{35v-20}{v-9}\)
\(\dfrac{x^2+11x+24}{6x^3+18x^2}\cdot\dfrac{6x^3+6x^2}{x^2+5x-24}\)
\(\dfrac{k-7}{k^2-k-12}\cdot\dfrac{7k^2-28k}{8k^2-56k}\)
\(\dfrac{9x^3+54x^2}{x^2+5x-14}\cdot\dfrac{x^2+5x-14}{10x^2}\)
\(\dfrac{n-7}{n^2-2n-35}\div\dfrac{9n+54}{10n+50}\)
\(\dfrac{7x^2-66x+80}{49x^2+7x-72}\div\dfrac{7x^2+39x-70}{49x^2+7x-72}\)
\(\dfrac{35n^2-12n-32}{49n^2-91n+40}\cdot\dfrac{7n^2+16n-15}{5n+4}\)
\(\dfrac{12x+24}{10x^2+34x+28}\cdot\dfrac{15x+21}{5}\)
\(\dfrac{x^2-1}{2x-4}\cdot\dfrac{x^2-4}{x^2-x-2}\div\dfrac{x^2+x-2}{3x-6}\)
\(\dfrac{x^2+3x+9}{x^2+x-12}\cdot\dfrac{x^2+2x-8}{x^3-27}\div\dfrac{x^2-4}{x^2-6x+9}\)
\(\dfrac{a^3+b^3}{a^2+3ab+2b^2}\cdot\dfrac{3a-6b}{3a^2-3ab+3b^2}\div\dfrac{a^2-4b^2}{a+2b}\)
\(\dfrac{x^2+3x-10}{x^2+6x+5}\cdot\dfrac{2x^2-x-3}{2x^2+x-6}\div\dfrac{8x+20}{6x+15}\)
Realizar la operación indicada y simplificar.
Dejar\(f(x)=\dfrac{5x^2+8x+3}{5x^2+7x+2}\) y\(g(x)=\dfrac{x^2-4x+3}{5x^2-2x-3}\). Encuentra y simplifica\((P\cdot R)(x)\).
Dejar\(f(x)=\dfrac{4x^2-21x+5}{4x^2-23x+15}\) y\(g(x)=\dfrac{x^2+5x+6}{4x^2+11x-3}\). Encuentra y simplifica\((P\cdot R)(x)\).
Dejar\(P(x)=\dfrac{3x^2-10x+8}{3x^2-4x-4}\) y\(R(x)=\dfrac{3x^2+8x-16}{x^2+5x+4}\). Encuentra y simplifica\((P\div R)(x)\).
Dejar\(P(x)=\dfrac{4x^2+19x-5}{4x^2+17x-15}\) y\(R(x)=\dfrac{4x^2-21x+5}{x^2-3x-10}\). Encuentra y simplifica\((P\div R)(x)\).