Multiplying and Dividing Rational Expressions

Multiplying: top times top, bottom times bottom

Multiplying rational expressions follows the same rule as 2/3 · 4/5 = 8/15: multiply numerators, multiply denominators. The smart order, though, is to factor first, cancel, then multiply — that keeps the numbers small and the answer already in lowest terms.

Multiply   (x^2 - 4)/(x^2 + 6x + 9)  ·  (x + 3)/(x - 2)

Factor every piece:
   (x - 2)(x + 2)        (x + 3)
  ----------------  ·  ---------
     (x + 3)(x + 3)      (x - 2)

Cancel (x - 2) and one (x + 3) across the product:

   (x + 2)
  ---------
   (x + 3)

Result:  (x + 2)/(x + 3),   x ≠ -3,  x ≠ 2

Dividing: multiply by the reciprocal

To divide, flip the divisor and multiply. The flipped fraction is the reciprocal of the expression — its multiplicative inverse. Just as 1/2 ÷ 3/4 = 1/2 · 4/3, dividing rational expressions turns into a multiplication you already know how to do.

Divide   (x^2 - 1)/(x + 4)  ÷  (x - 1)/(x^2 + 4x)

Flip the second fraction and multiply:
   (x^2 - 1)/(x + 4)  ·  (x^2 + 4x)/(x - 1)

Factor:
   (x - 1)(x + 1)        x(x + 4)
  ----------------  ·  -----------
     (x + 4)              (x - 1)

Cancel (x - 1) and (x + 4):
   (x + 1) · x  =  x(x + 1)  =  x^2 + x

Result:  x^2 + x,   x ≠ -4, x ≠ 0, x ≠ 1

The one workflow for both

1. If it is a division, flip the second fraction to a reciprocal and change ÷ to ·.
2. Factor every numerator and every denominator completely.
3. Cancel any factor on a top against the same factor on a bottom.
4. Multiply what is left; note excluded values from every denominator that appeared.