Necesito ayuda con este ejercicio.
Adjuntos:
![](https://es-static.z-dn.net/files/d96/c6c9c471d0235670d9f3d6f0168dc62a.jpg)
CristianFRC:
Listo jeje... se me había ido la respuesta incompleta pero ya está bien! :D Saludos
Respuestas
Respuesta dada por:
1
Primero factorizamos cada expresión:
![15x^2+7x-2=15(x^2+ \frac{7}{15} x- \frac{2}{15} )=15(x+ \frac{2}{3})(x- \frac{1}{5} ) \\ 25x^3-x=25x(x^2- \frac{1}{25} )=25x(x+ \frac{1}{5} )(x- \frac{1}{5} )\\ 6x^2+13x+6=6(x^2+ \frac{13}{6}x+1)=6(x+ \frac{3}{2} )(x+ \frac{2}{3} ) \\ 25x^2+10x+1=25(x^2+ \frac{10}{25}x+ \frac{1}{25} )=25(x+ \frac{1}{5} )^2=25(x+ \frac{1}{5} )(x+ \frac{1}{5} ) 15x^2+7x-2=15(x^2+ \frac{7}{15} x- \frac{2}{15} )=15(x+ \frac{2}{3})(x- \frac{1}{5} ) \\ 25x^3-x=25x(x^2- \frac{1}{25} )=25x(x+ \frac{1}{5} )(x- \frac{1}{5} )\\ 6x^2+13x+6=6(x^2+ \frac{13}{6}x+1)=6(x+ \frac{3}{2} )(x+ \frac{2}{3} ) \\ 25x^2+10x+1=25(x^2+ \frac{10}{25}x+ \frac{1}{25} )=25(x+ \frac{1}{5} )^2=25(x+ \frac{1}{5} )(x+ \frac{1}{5} )](https://tex.z-dn.net/?f=15x%5E2%2B7x-2%3D15%28x%5E2%2B+%5Cfrac%7B7%7D%7B15%7D+x-+%5Cfrac%7B2%7D%7B15%7D+%29%3D15%28x%2B+%5Cfrac%7B2%7D%7B3%7D%29%28x-+%5Cfrac%7B1%7D%7B5%7D+%29++%5C%5C+25x%5E3-x%3D25x%28x%5E2-+%5Cfrac%7B1%7D%7B25%7D+%29%3D25x%28x%2B+%5Cfrac%7B1%7D%7B5%7D+%29%28x-+%5Cfrac%7B1%7D%7B5%7D+%29%5C%5C+6x%5E2%2B13x%2B6%3D6%28x%5E2%2B+%5Cfrac%7B13%7D%7B6%7Dx%2B1%29%3D6%28x%2B+%5Cfrac%7B3%7D%7B2%7D+%29%28x%2B+%5Cfrac%7B2%7D%7B3%7D+%29+%5C%5C+25x%5E2%2B10x%2B1%3D25%28x%5E2%2B+%5Cfrac%7B10%7D%7B25%7Dx%2B++%5Cfrac%7B1%7D%7B25%7D+%29%3D25%28x%2B+%5Cfrac%7B1%7D%7B5%7D+%29%5E2%3D25%28x%2B+%5Cfrac%7B1%7D%7B5%7D+%29%28x%2B+%5Cfrac%7B1%7D%7B5%7D+%29)
Entonces reemplazamos en las dos primeras expresiones:
![\frac{15x^2+7x-2}{25x^3-x} = \frac{15(x+ \frac{2}{3} )(x- \frac{1}{5} )}{25x(x+ \frac{1}{5} )(x-\frac{1}{5})} = \frac{15(x+ \frac{2}{3} )}{25x(x+ \frac{1}{5} )} \frac{15x^2+7x-2}{25x^3-x} = \frac{15(x+ \frac{2}{3} )(x- \frac{1}{5} )}{25x(x+ \frac{1}{5} )(x-\frac{1}{5})} = \frac{15(x+ \frac{2}{3} )}{25x(x+ \frac{1}{5} )}](https://tex.z-dn.net/?f=+%5Cfrac%7B15x%5E2%2B7x-2%7D%7B25x%5E3-x%7D+%3D+%5Cfrac%7B15%28x%2B+%5Cfrac%7B2%7D%7B3%7D+%29%28x-+%5Cfrac%7B1%7D%7B5%7D+%29%7D%7B25x%28x%2B+%5Cfrac%7B1%7D%7B5%7D+%29%28x-%5Cfrac%7B1%7D%7B5%7D%29%7D+%3D+%5Cfrac%7B15%28x%2B+%5Cfrac%7B2%7D%7B3%7D+%29%7D%7B25x%28x%2B+%5Cfrac%7B1%7D%7B5%7D+%29%7D+)
![\frac{6x^2+13x+6}{25x^2+10x+1} = \frac{6(x+ \frac{3}{2})(x+\frac{2}{3}) }{25(x+\frac{1}{5})(x+\frac{1}{5})} \frac{6x^2+13x+6}{25x^2+10x+1} = \frac{6(x+ \frac{3}{2})(x+\frac{2}{3}) }{25(x+\frac{1}{5})(x+\frac{1}{5})}](https://tex.z-dn.net/?f=+%5Cfrac%7B6x%5E2%2B13x%2B6%7D%7B25x%5E2%2B10x%2B1%7D+%3D+%5Cfrac%7B6%28x%2B+%5Cfrac%7B3%7D%7B2%7D%29%28x%2B%5Cfrac%7B2%7D%7B3%7D%29+%7D%7B25%28x%2B%5Cfrac%7B1%7D%7B5%7D%29%28x%2B%5Cfrac%7B1%7D%7B5%7D%29%7D+)
(*Esta segunda no se puede reducir más, procedemos a dividir estas dos expresiones)
![\frac{\frac{15(x+ \frac{2}{3} )}{25x(x+ \frac{1}{5} )}}{\frac{6(x+ \frac{3}{2})(x+\frac{2}{3}) }{25(x+\frac{1}{5})(x+\frac{1}{5})}} = \frac{15(x+ \frac{2}{3} )*25(x+\frac{1}{5})(x+\frac{1}{5})}{25x(x+ \frac{1}{5} )*6(x+ \frac{3}{2})(x+\frac{2}{3})} = \frac{15(x+ \frac{1}{5}) }{6x(x+ \frac{3}{2} } = \frac{15x+3}{6x^2+9x} = \frac{5x+1}{2x^2+3x} \frac{\frac{15(x+ \frac{2}{3} )}{25x(x+ \frac{1}{5} )}}{\frac{6(x+ \frac{3}{2})(x+\frac{2}{3}) }{25(x+\frac{1}{5})(x+\frac{1}{5})}} = \frac{15(x+ \frac{2}{3} )*25(x+\frac{1}{5})(x+\frac{1}{5})}{25x(x+ \frac{1}{5} )*6(x+ \frac{3}{2})(x+\frac{2}{3})} = \frac{15(x+ \frac{1}{5}) }{6x(x+ \frac{3}{2} } = \frac{15x+3}{6x^2+9x} = \frac{5x+1}{2x^2+3x}](https://tex.z-dn.net/?f=+%5Cfrac%7B%5Cfrac%7B15%28x%2B+%5Cfrac%7B2%7D%7B3%7D+%29%7D%7B25x%28x%2B+%5Cfrac%7B1%7D%7B5%7D+%29%7D%7D%7B%5Cfrac%7B6%28x%2B+%5Cfrac%7B3%7D%7B2%7D%29%28x%2B%5Cfrac%7B2%7D%7B3%7D%29+%7D%7B25%28x%2B%5Cfrac%7B1%7D%7B5%7D%29%28x%2B%5Cfrac%7B1%7D%7B5%7D%29%7D%7D+%3D+%5Cfrac%7B15%28x%2B+%5Cfrac%7B2%7D%7B3%7D+%29%2A25%28x%2B%5Cfrac%7B1%7D%7B5%7D%29%28x%2B%5Cfrac%7B1%7D%7B5%7D%29%7D%7B25x%28x%2B+%5Cfrac%7B1%7D%7B5%7D+%29%2A6%28x%2B+%5Cfrac%7B3%7D%7B2%7D%29%28x%2B%5Cfrac%7B2%7D%7B3%7D%29%7D+%3D+%5Cfrac%7B15%28x%2B+%5Cfrac%7B1%7D%7B5%7D%29+%7D%7B6x%28x%2B+%5Cfrac%7B3%7D%7B2%7D+%7D+%3D+%5Cfrac%7B15x%2B3%7D%7B6x%5E2%2B9x%7D+%3D+%5Cfrac%7B5x%2B1%7D%7B2x%5E2%2B3x%7D+)
Entonces reemplazamos en las dos primeras expresiones:
(*Esta segunda no se puede reducir más, procedemos a dividir estas dos expresiones)
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