How do you simplify #(30x^2+ 2x) /( x^2+x-2 ) *( (x+2)(x-2))/(15x^3-30x^2)#?

Question

Julia Adams · Accepted Answer

#(2(15x+1))/(15x(x-1)#

Your starting expression is

#(30x^2 + 2x)/(x^2 + x - 2) * ((x+2)(x-2))/(15x^3 - 30x^2)#

Your first step is to try and simplify the numerators and denominators as much as possible by factoring them.

#30x^2 + 2x = 2x(15x + 1)#

#x^2 + x - 2 = x^2 + 2x - x - 2#

#=x(x+2) - (x+2)#

#=(x+2)(x-1)#

and

#15x^3 - 30x^2 = 15x^2(x-2)#

The expression can thus be rewritten as

#(2x(15x+1))/((x+2)(x-1)) * ((x+2)(x-2))/(15x^2(x-2))#

Notice that the expressions that can be found both in the numerator, and in the denominator cancel out to give

#(2x(15x+1))/(color(red)(cancel(color(black)((x+2))))(x-1)) * (color(red)(cancel(color(black)((x+2))))color(blue)(cancel(color(black)((x-2)))))/(15x^2color(blue)(cancel(color(black)((x-2)))))#

You are left with

#(2color(purple)(cancel(color(black)(x)))(15x+1))/(x-1) * 1/(15x * color(purple)(cancel(color(black)(x)))) = color(green)( (2(15x+1))/(15x(x-1))#

Eva Abramson · Answer

undefined To simplify the expression (30x^2+ 2x) /( x^2+x-2 ) *( (x+2)(x-2))/(15x^3-30x^2), we can follow these steps:

1. Factorize the denominators:
   - x^2+x-2 can be factored as (x+2)(x-1)
   - 15x^3-30x^2 can be factored as 15x^2(x-2)

2. Simplify the expression by canceling out common factors:
   - (30x^2+ 2x) / (x+2)(x-1) * (x+2)(x-2) / 15x^2(x-2)
   - Cancel out the common factors: (x-2) and (x+2)
   - Simplified expression: (30x^2+ 2x) / 15x^2(x-1)

Therefore, the simplified expression is (30x^2+ 2x) / 15x^2(x-1).