www.VirtualNerd.com

How Do You Multiply Two Rational Expressions?

Multiply and simplify the following the rational expressions shown on the right.

Summary

  1. Factor the numerator and denominator of each fraction and cancel out like terms
  2. The first numerator factors into 4 times (x+2)
  3. The first denominator factors into (x+5)(x-5)
  4. The second numerator, x-5, cannot be factored any further
  5. The second denominator factors into 5 times (x+2)
  6. There is an (x+2) in both the top and bottom, so those cancel
  7. There is an (x-5) in both the top and bottom, so those also cancel

Notes

    1. Our first rational expression is 4x+8 divided by x2-25
    1. 4x+8 is the same as 4(x+2)
    2. 4 is the greatest common factor of 4x and 8
    1. x2 is the same as (x+5)(x-5)
    2. x2-25 is a difference of squares
    1. Our second rational expression is x-5 over 5x-10
    1. 5x+10 is the same as 5(x+2)
    2. 5 is the greatest common factor of 5x and 10
    1. We can rearrange the numerator and denominators so that (x+2) is being divided by (x+2), which is always 1
    1. We can rearrange the numerator and denominators so that (x+2) is being divided by (x+2), which is always 1
    1. We can rearrange the numerator and denominators so that (x-5) is being divided by (x-5), which is always 1
    1. We can rearrange the numerator and denominators so that (x-5) is being divided by (x-5), which is always 1
    1. There is no 4 in the denominator and neither an (x+5) nor 5 in the numerator, so nothing can be canceled out