Divide Rational Expressions

Dividing Rational Expressions

When you divide rational expressions, treat the problem like multiplying by the reciprocal of the second fraction. The main goal is to simplify the result as much as possible.

1) Rewrite the division as multiplication

If you have [ \frac{A}{B} \div \frac{C}{D}, ] change it to [ \frac{A}{B} \times \frac{D}{C}. ] This is the key step in every problem.

2) Factor completely

Before simplifying, factor every numerator and denominator as much as possible. Look for common factors in polynomials, such as binomials or differences of squares.

3) Cancel common factors

After rewriting as multiplication, cancel factors that appear in both a numerator and a denominator. Cancel factors, not terms added or subtracted.

4) Multiply what remains

Multiply the remaining numerators together and the remaining denominators together. Then simplify the final fraction if possible.

5) Check restrictions

Any value that makes an original denominator zero is not allowed. Also, the expression you divide by cannot be zero, so its numerator must not be zero either.

Quick check

Your final answer should be simplified, with no common factors left between numerator and denominator, and it should respect all domain restrictions from the original expressions.