1Cademy - Example of Dividing the Rational Functions $$f(x) = \frac{2x^2}{x^2 - 8x}$$ and $$g(x) = \frac{8x^2 + 24x}{x^2 + x

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How to Divide Rational Expressions

Example

Example of Dividing the Rational Functions $f(x) = \frac{2x^2}{x^2 - 8x}$ and $g(x) = \frac{8x^2 + 24x}{x^2 + x - 6}$

To find the quotient $R(x) = \frac{f(x)}{g(x)}$ for the rational functions $f(x) = \frac{2x^2}{x^2 - 8x}$ and $g(x) = \frac{8x^2 + 24x}{x^2 + x - 6}$ , apply the procedure for dividing rational expressions:

Step 1. Substitute the polynomial expressions for $f(x)$ and $g(x)$ : $R(x) = \frac{\frac{2x^2}{x^2 - 8x}}{\frac{8x^2 + 24x}{x^2 + x - 6}}$

Step 2. Convert the division to multiplication by taking the reciprocal of $g(x)$ : $R(x) = \frac{2x^2}{x^2 - 8x} \cdot \frac{x^2 + x - 6}{8x^2 + 24x}$

Step 3. Factor the numerators and denominators completely, then multiply: $R(x) = \frac{2x^2(x + 3)(x - 2)}{x(x - 8) \cdot 8x(x + 3)}$

Expanding the numerical factors to reveal all common parts: $R(x) = \frac{2 \cdot x \cdot x \cdot (x + 3)(x - 2)}{x(x - 8) \cdot 2 \cdot 4 \cdot x \cdot (x + 3)}$

Step 4. Simplify by dividing out the common factors of $2$ , $x$ , $x$ , and $(x + 3)$ : $R(x) = \frac{x - 2}{4(x - 8)}$

Updated 2026-04-30

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OpenStax Intermediate Algebra

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