Question

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

Answer 1

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

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->` nothing to do;
  **Denominator: `4x-8->` can factor a `4`, obtaining `4(x-2)`.

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

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