Question

Necesito ayuda con este ejercicio.

Answer 1

Primero factorizamos cada expresión:

$15x^2+7x-2=15(x^2+ \frac{7}{15} x- \frac{2}{15} )=15(x+ \frac{2}{3})(x- \frac{1}{5} ) \\ 25x^3-x=25x(x^2- \frac{1}{25} )=25x(x+ \frac{1}{5} )(x- \frac{1}{5} )\\ 6x^2+13x+6=6(x^2+ \frac{13}{6}x+1)=6(x+ \frac{3}{2} )(x+ \frac{2}{3} ) \\ 25x^2+10x+1=25(x^2+ \frac{10}{25}x+ \frac{1}{25} )=25(x+ \frac{1}{5} )^2=25(x+ \frac{1}{5} )(x+ \frac{1}{5} )$

Entonces reemplazamos en las dos primeras expresiones:

$\frac{15x^2+7x-2}{25x^3-x} = \frac{15(x+ \frac{2}{3} )(x- \frac{1}{5} )}{25x(x+ \frac{1}{5} )(x-\frac{1}{5})} = \frac{15(x+ \frac{2}{3} )}{25x(x+ \frac{1}{5} )}$

$\frac{6x^2+13x+6}{25x^2+10x+1} = \frac{6(x+ \frac{3}{2})(x+\frac{2}{3}) }{25(x+\frac{1}{5})(x+\frac{1}{5})}$
(*Esta segunda no se puede reducir más, procedemos a dividir estas dos expresiones)

$\frac{\frac{15(x+ \frac{2}{3} )}{25x(x+ \frac{1}{5} )}}{\frac{6(x+ \frac{3}{2})(x+\frac{2}{3}) }{25(x+\frac{1}{5})(x+\frac{1}{5})}} = \frac{15(x+ \frac{2}{3} )*25(x+\frac{1}{5})(x+\frac{1}{5})}{25x(x+ \frac{1}{5} )*6(x+ \frac{3}{2})(x+\frac{2}{3})} = \frac{15(x+ \frac{1}{5}) }{6x(x+ \frac{3}{2} } = \frac{15x+3}{6x^2+9x} = \frac{5x+1}{2x^2+3x}$