How do you simplify #[(2x^3+3x^2-2x)/( 3x-15) ÷ (2x^3-x^2)/ (x^2-3x-10)] * (5x^2-10x)/(3x^2+12x+12)#?

Question

Ethan Adkins · Accepted Answer

#E = 5/18 * (x-2)#

Start by writing out your initial expression

#E = (2x^3 + 3x^2 - 2x)/(3x-15) * (x^2 - 3x - 10)/(2x^3 - x^2) * (5x^2 - 10x)/(3x^2 + 12x + 12)#

You can factor the quadratics by using the sum/product technique to get

#x^2 - 3x - 10 = x^2 + 2x - 5x -10 = (x-5)(x+2)#

and

#3(x^2 + 4x + 4) = 3(x^2 + 2x + 2x + 4) = 3(x+2)(x+2)#

Your expression can thus be written as

#E = (color(blue)(cancel(color(black)(x)))(2x^2 + 3x -2))/(3color(red)(cancel(color(black)((x-5))))) * (color(red)(cancel(color(black)((x-5))))color(purple)(cancel(color(black)((x+2)))))/(x^(color(blue)(cancel(color(black)(2))))(2x-1)) * (5color(blue)(cancel(color(black)(x)))(x-2))/(3color(purple)(cancel(color(black)((x+2))))(x+2))#

#E = (2x^2 + 3x - 2)/3 * 1/(2x-1) * (5(x-2))/(3(x+2))#

The remaining quadratic can be factored by using the quadratic formula

#color(blue)(x_(1,2) = (-b +- sqrt(b^2 - 4ac))/(2a))#

#x_(1,2) = (-3 +- sqrt(3^2 - 4 * 2 * (-2)))/(2 *2)#

#x_(1,2) = (-3 +- 5)/4 = { (x_1 = (-3-5)/4 = -2), (x_2 = (-3 + 5)/4 = 1/2) :}#

This means that you have

#E = (color(red)(cancel(color(black)((x+2))))(x-1/2))/3 * 1/(2x-1) * (5(x-2))/(3color(red)(cancel(color(black)((x+2)))))#

#E = 5/9 * ((x-2)color(red)(cancel(color(black)((x-1/2)))))/(2color(red)(cancel(color(black)((x-1/2))))) = color(green)(5/18 * (x-2)#

Hazel Anglin · Answer

undefined To simplify the expression [(2x^3+3x^2-2x)/(3x-15) ÷ (2x^3-x^2)/(x^2-3x-10)] * (5x^2-10x)/(3x^2+12x+12), we can follow these steps:

1. Factorize the numerator and denominator of each fraction:
   - Numerator of the first fraction: 2x^3+3x^2-2x = x(2x^2+3x-2) = x(2x-1)(x+2)
   - Denominator of the first fraction: 3x-15 = 3(x-5)
   - Numerator of the second fraction: 2x^3-x^2 = x^2(2x-1)
   - Denominator of the second fraction: x^2-3x-10 = (x-5)(x+2)
   - Numerator of the third fraction: 5x^2-10x = 5x(x-2)
   - Denominator of the third fraction: 3x^2+12x+12 = 3(x^2+4x+4)

2. Rewrite the expression with the factored forms:
   [(x(2x-1)(x+2))/(3(x-5))] ÷ [(x^2(2x-1))/((x-5)(x+2))] * [(5x(x-2))/(3(x^2+4x+4))]

3. Simplify by canceling out common factors:
   [(x)/(3)] * [(5(x-2))/(x^2+4x+4)]
   (5x(x-2))/(3(x^2+4x+4))

Therefore, the simplified expression is (5x(x-2))/(3(x^2+4x+4)).