1.8: Fracciones Complejas
- Page ID
- 112013
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1.8 Fracciones
Complejas Las fracciones complejas implican simplificar una expresión racional que tiene un numerador y/o denominador complicado
Ejemplo
Simplificar.
\ [
\ frac {3+\ frac {x} {x+2}} {1-\ frac {x+3} {x-1}}
\]
Hay una variedad de formas de abordar este problema. Una de las formas más sencillas de simplificar la expresión anterior es crear denominadores comunes para el numerador y el denominador de manera que cada uno sea una sola expresión fraccionaria:
\ [
\ begin {aligned}
\ frac {3+\ frac {x} {x+2}} {1-\ frac {x+3} {x-1}} &=\ frac {\ frac {\ frac {3} {1} *\ frac {x+2} {x+2} +\ frac {x} {x+2}} {\ frac {1} {1} *\ frac {x-1} {x-1} -\ frac {x+3} {x-1}}\\
&=\ frac {\ izquierda (\ frac {3 x+6+x} {x+2}\ derecha)} {\ left (\ frac {x-1- (x+3)} {x-1}\ derecha)}
\ end {alineado}
\]
\(=\frac{\left(\frac{4 x+6}{x+2}\right)}{\left(\frac{-4}{x-1}\right)} \quad\) (Ahora esto es un problema de división)
\ [
\ begin {array} {l}
=\ frac {4 x+6} {x+2} *\ frac {x-1} {-4} =\ frac {2 (2 x+3)} {x+2} *\ frac {x-1} {-4}\\
=\ frac {\ cancel {2} (2 x+3)} {x+2} *\ frac {x-1} {cancelar\ {-4} (-2)} =\ frac {(2 x+3) (x-1)} {-2 (x+2)}
\ end {array}
\]
La simplificación de fracciones complejas utiliza todos los conceptos anteriores sobre expresiones racionales que hemos cubierto en este capítulo.
Ejemplo
Simplificar.
\ [
\ frac {x-\ frac {x} {x+3}} {1+\ frac {2} {x}}
\]
\(\frac{x-\frac{x}{x+3}}{1+\frac{2}{x}}=\frac{\frac{x}{1} * \frac{x+3}{x+3}-\frac{x}{x+3}}{\frac{1}{1} * \frac{x}{x}+\frac{2}{x}} \quad\) creando denominadores comunes
\ [
\ begin {array} {l}
=\ frac {\ left (\ frac {x (x+3) -x} {x+3}\ right)} {\ left (\ frac {x+2} {x}\ derecha)}\\
=\ frac {\ izquierda ( \ frac {x^ {2} +3 x-x} {x+3}\ derecha)} {\ izquierda (\ frac {x+2} {x}\ derecha)} =\ frac {\ izquierda (\ frac {x^ {2} +2 x} {x+3}\ derecha)} {\ izquierda (\ frac {x+2} {x}\ derecha)}\ texto {fracciones dividiendo}\
=\ frac {x^ {2} +2 x} {x+3} *\ frac {x} {x+2} =\ frac {x (x+2)} {x+3} *\ frac {x} {x+2}
\ final {matriz}
\]
\(=\frac{x\cancel{(x+2)}}{x+3} * \frac{x}{\cancel{x+2}} \quad\)factor y cancelar para reducir a los términos más bajos
\ [
=\ frac {x^ {2}} {x+3}
\]
Ejercicios 1.8
Simplificar cada fracción compleja. Exprese su respuesta en los términos más bajos.
1)\(\frac{1}{\left(x+\frac{y}{2}\right)}\)
2)\(\frac{\left(\frac{1}{x}+\frac{1}{y}\right)}{\left(\frac{y}{x}-\frac{x}{y}\right)}\)
3)\(\frac{\left(1+\frac{m}{n}\right)}{\left(1-\frac{n^{2}}{m^{2}}\right)}\)
4)\(\frac{\left(\frac{1}{x}-\frac{1}{y}\right)}{\left(\frac{1}{x^{2}}-\frac{1}{y^{2}}\right)}\)
5)\(\frac{\left(\frac{x}{y}-\frac{x-y}{x+y}\right)}{\left(\frac{y}{x}+\frac{x-y}{x+y}\right)}\)
6)\(\frac{\left(\frac{7}{a+1}-\frac{3}{a}\right)}{\left(\frac{3}{a}+\frac{1}{a-1}\right)}\)
7)\(\frac{\left(x-\frac{1}{2 x+1}\right)}{\left(1-\frac{2}{2 x+1}\right)}\)
8)\(\frac{\left(\frac{1}{2 x-2}-\frac{1}{x}\right)}{\left(\frac{2}{x}-\frac{1}{x-1}\right)}\)
9)\(\frac{\left(x+\frac{4}{x+4}\right)}{\left(x-\frac{4 x+4}{x+4}\right)}\)
10)\(\frac{\left(x-\frac{x+6}{x+2}\right)}{\left(x-\frac{4 x+15}{x+2}\right)}\)
11)\(\frac{\left(\frac{1}{x+2}-\frac{1}{x-3}\right)}{\left(1+\frac{1}{x^{2}-x-6}\right)}\)
\(\frac{\left(1-\frac{1}{x+1}\right)}{\left(1+\frac{1}{x-1}\right)}\)
13)\(\frac{\left(\frac{1}{a-b}-\frac{3}{a+b}\right)}{\left(\frac{2}{b-a}+\frac{4}{b+a}\right)}\)
14)\(\frac{\left(\frac{3}{y^{2}-4}\right)}{\left(\frac{1}{y+2}-\frac{1}{y-2}\right)}\)
\(\frac{\left(n+2-\frac{5}{n-2}\right)}{\left(1-\frac{1}{(n-2)^{2}}\right)}\)
16)\(\frac{\left(4+\frac{1}{x+1}\right)}{\left(16-\frac{1}{(x+1)^{2}}\right)}\)
17)\(\frac{\left(2+\frac{x-2}{1-x^{2}}\right)}{\left(2-\frac{3}{x+1}\right)}\)
18)\(\frac{\left(\frac{1}{2 x-1}-\frac{1}{2 x+1}\right)}{\left(4-\frac{1}{x^{2}}\right)}\)