Section 9.2: Multiplying & Dividing Rational Expressions

To multiply rational expressions, multiply numerators together and denominators together, then simplify by factoring and canceling common terms.

To divide rational expressions, multiply by the reciprocal of the divisor and then simplify.

Example 1

Multiply: \( \frac{x^2 - 4}{x^2 - x - 6} \times \frac{x^2 - 9}{x^2 - 2x - 8} \)

Factor all polynomials:

\( x^2 - 4 = (x-2)(x+2) \)

\( x^2 - x - 6 = (x-3)(x+2) \)

\( x^2 - 9 = (x-3)(x+3) \)

\( x^2 - 2x - 8 = (x-4)(x+2) \)

After canceling: \( \frac{(x-2)(x+3)}{(x-4)(x+2)} \), restrictions: \( x \neq -2, 3, 4 \).

Divide: \( \frac{2x^2}{x^2 - 9} \div \frac{4x}{x^2 - 3x} \)

Rewrite as multiplication by reciprocal:

\( \frac{2x^2}{x^2 - 9} \times \frac{x^2 - 3x}{4x} \)

Factor: \( x^2 - 9 = (x-3)(x+3), \; x^2 - 3x = x(x-3) \)

So expression becomes: \( \frac{2x^2 \cdot x(x-3)}{4x (x-3)(x+3)} \)

Cancel common factors: result \( \frac{x^2}{2(x+3)} \), restrictions: \( x \neq -3, 0, 3 \).

\( \frac{x^2 - 1}{x^2 - 4} \times \frac{x+2}{x-1} \)
\( \frac{y^2 - 9}{y^2 - y - 6} \div \frac{y+3}{y-2} \)
\( \frac{3m^2}{m^2 - 16} \times \frac{m-4}{6m} \)
\( \frac{a^2 - 25}{2a} \div \frac{a-5}{4} \)
\( \frac{p^2 + 3p}{p^2 - 9} \times \frac{p+3}{p} \)