Multiplying and Dividing Rational Expressions

How to multiply rational expressions

The strategy: factor everything, then cancel common factors between any numerator and any denominator before multiplying.

Step-by-step: $\dfrac{x^2-4}{x+3} \cdot \dfrac{x+3}{x-2}$

Step 1 — Factor all numerators and denominators:

$\frac{(x-2)(x+2)}{(x+3)} \cdot \frac{(x+3)}{(x-2)}$

Step 2 — Cancel common factors (a factor in any numerator can cancel with the same factor in any denominator):

$(x+3)$ appears in the first denominator and second numerator — cancel!

$(x-2)$ appears in the first numerator and second denominator — cancel!

Step 3 — Multiply remaining factors:

$= \frac{(x+2)}{1} = x + 2$

Step 4 — State domain restrictions: $x \neq -3$ and $x \neq 2$ (from the original denominators).

Check: At $x = 1$: Original = $\frac{1-4}{1+3} \cdot \frac{1+3}{1-2} = \frac{-3}{4} \cdot \frac{4}{-1} = 3$. Simplified: $1 + 2 = 3$ ✓.

Division of rational expressions

To divide, flip the second fraction and multiply: $\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C}$. Then factor and cancel as above.

Try it in the visualization

Watch the factor-cancel animation. Common factors highlight and cross out. The graphs show the original product equals the simplified result (except at excluded points).