9.2: Multiplicar expresiones racionales
- Page ID
- 112583
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Para multiplicar expresiones racionales, multiplicar las expresiones del numerador y multiplicar las expresiones del denominador. Entonces, si es posible, simplificar factorizando tanto el numerador como el denominador y eliminando factores comunes.
Trate de usar factor agrupando cuando trabaje con un polinomio\(4\) -term.
Nota: ¡No se necesitan denominadores comunes a la hora de multiplicar expresiones racionales!
Multiplicar las expresiones racionales:
- \(\dfrac{10t}{6t − 12} ∗ \dfrac{20t − 40}{30t^3}\)
- \(\dfrac{7x + 14}{ 2x^2 − 8} ∗ \dfrac{x^2 + 3x − 10}{14x + 21}\)
- \(\dfrac{3x^3 − 24}{2x^2 − 14x + 20} ∗ \dfrac{4x^3 − 20x^2 + 3x − 15}{3x^2 + 6x + 12}\)
Solución
- \(\begin{array} &&\dfrac{10t}{6t − 12} ∗ \dfrac{20t − 40}{30t^3} &\text{Example problem} \\ &\dfrac{10t}{6t − 12} ∗ \dfrac{20(t − 2)}{30t^3} &\text{Factor all numerators and denominators} \\ &\dfrac{(2)(5)(t)(2)(2)(5)(t − 2)}{(2)(3)(t − 2)(2)(3)(5)(t)(t^2)} &\text{Factor numbers into prime factors and write with one division bar} \\ &\dfrac{\cancel{(2)}(5)\cancel{(t)}(2)(2)\cancel{(5)}\cancel{(t − 2)}}{\cancel{(2)}(3)\cancel{(t − 2)}(2)(3)\cancel{(5)}\cancel{(t)}(t^2)} &\text{Remove common factors} \\ &\dfrac{(5)(2)(2)}{(3)(2)(3)(t^2)} &\text{Simplify} \\ &\dfrac{20}{18t^2} &\text{Final answer (it is good practice to show final answer as factored as possible)} \end{array} \)
- \(\begin{array} &&\dfrac{7x + 14}{2x^2 − 8} ∗ \dfrac{x^2 + 3x − 10}{14x + 21} &\text{Example problem} \\ &\dfrac{7(x + 2)}{2(x^2 − 4)} ∗ \dfrac{(x + 5)(x − 2)}{7(2x + 3)} &\text{Factor all numerators and denominators} \\ &\dfrac{7(x + 2)(x + 5)(x − 2)}{2(x + 2)(x − 2)(7)(2x + 3)} &\text{Further factor algebraic expressions and numbers into prime factors and write with one division bar} \\ &\dfrac{\cancel{7}\cancel{(x + 2)}(x + 5)(x − 2)}{2\cancel{(x + 2)}(x − 2)\cancel{(7)}(2x + 3)} &\text{Remove common factors} \\ &\dfrac{(x + 5)(x − 2)}{2(x − 2)(2x + 3)} &\text{Final answer} \end{array}\)
- \(\begin{array} &&\dfrac{3x^3 − 24}{2x^2 − 14x + 20} ∗ \dfrac{4x^3 − 20x^2 + 3x − 15}{3x^2 + 6x + 12} &\text{Example problem} \\ &\dfrac{3(x^3 − 8)}{2(x^2 − 7x + 10)} ∗ \dfrac{4x^2 (x − 5) + 3(x − 5)}{3(x^2 + 2x + 4)} &\text{Factor all numerators and denominators. Use factor by grouping for the \(4\)-term polinomio.}\\ &\ dfrac {3 (x − 2) (x^2 + 2x + 4)} {2 (x − 5) (x − 2)} ∗\ dfrac {(4x^2 + 3) (x − 5)} {3 (x + 2) (x + 2)} &\ text {expresiones algebraicas de factores adicionales}\\ & dfrac ac {3 (x − 2) (x^2 + 2x + 4) (4x^2 + 3) (x − 5)} {2 (x − 5) (x − 2) (3) (x + 2) (x + 2)} &\ text {Factor adicional algebraico expresiones y números en factores primos y escribir con una barra de división}\\ &\ dfrac {\ cancel {3}\ cancel {(x − 2)} (x^2 + 2x + 4) (4x^2 + 3)\ cancel {(x − 5)}} {2\ cancel {(x − 5)}\ cancel {(x − 2)}\ cancel {(3)} (x + 2) (x + 2)} &\ text {Eliminar factores comunes}\\ &\ dfrac {(x^2 + 2x + 4) (4x^2 + 3)} {2 (x + 2) (x + 2)} & amp;\ text {Respuesta final}\\ &\ dfrac {(x^2 + 2x + 4) (4x^2 + 3)} {2 (x + 2) ^2} &\ text {La respuesta final también se puede escribir en esta forma}\ end {array}\)
Multiplicar las expresiones racionales:
- \(\dfrac{x^2 + 4x + 3}{2x^2 − x − 10} ∗ \dfrac{2x^2 + 4x^3}{x^2 + 3x} ∗ \dfrac{x}{x^2 + 3x + 2}\)
- \(\dfrac{x^2 + 2x − 15}{x^2 − 4x − 45} ∗ \dfrac{x^2 − 5x − 36}{x^2 + x − 12}\)
- \(\dfrac{x^2 + 3x − 40}{x^2 + 2x − 35} ∗ \dfrac{x^2 + 3x − 18}{4x^2 − 5x − 32x + 40}\)