How do you simplify the rational expression: (x+2)/(4x-8)(3x-9)/(x+4)(2x-21)/(x^2-x-6)x+24x83x9x+42x21x2x6?

1 Answer
Oct 14, 2015

(6x-63)/(4 x^2+8 x-32)6x634x2+8x32

Explanation:

The idea is to factor out numbers is the first-degree polynomial (if possible), and to factor the highest-degree, finding their roots (if possible). So, let's work separately on the three pieces:

  • First fraction:
    Numerator: x+2->x+2 nothing to do;
    **Denominator: 4x-8->4x8 can factor a 44, obtaining 4(x-2)4(x2).

  • Second fraction:
    Numerator: 3x-9->3x9 can factor a 33, obtaining 3(x-3)3(x3);
    **Denominator: x+4->x+4 nothing to do.

  • First fraction:
    Numerator: 2x-21->2x21 nothing to do;
    **Denominator: x^2-x-6->x2x6 its roots are 33 and -22, so we can write it as (x-3)(x+2)(x3)(x+2).

Writing back the whole expression with this changes gives

color(red)(cancel(x+2))/(4(x-2)) * (3color(blue)cancel((x-3)))/(x+4) * (2x-21)/(color(blue)cancel((x-3))color(red)cancel((x+2)))