1Cademy - Example of Dividing the Rational Functions $$f(x) = \frac{3x^2}{x^2 - 4x}$$ and $$g(x) = \frac{9x^2 - 45x}{x^2

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How to Divide Rational Expressions

Example

Example of Dividing the Rational Functions $f(x) = \frac{3x^2}{x^2 - 4x}$ and $g(x) = \frac{9x^2 - 45x}{x^2 - 7x + 10}$

To find the quotient $R(x) = \frac{f(x)}{g(x)}$ for the rational functions $f(x) = \frac{3x^2}{x^2 - 4x}$ and $g(x) = \frac{9x^2 - 45x}{x^2 - 7x + 10}$ , follow the procedure for dividing rational expressions:

Step 1. Substitute the given functions into the division format: $R(x) = \frac{\frac{3x^2}{x^2 - 4x}}{\frac{9x^2 - 45x}{x^2 - 7x + 10}}$

Step 2. Rewrite the division as the product of $f(x)$ and the reciprocal of $g(x)$ : $R(x) = \frac{3x^2}{x^2 - 4x} \cdot \frac{x^2 - 7x + 10}{9x^2 - 45x}$

Step 3. Factor the numerators and denominators completely, then multiply: $R(x) = \frac{3 \cdot x \cdot x \cdot (x - 5)(x - 2)}{x(x - 4) \cdot 3 \cdot 3 \cdot x \cdot (x - 5)}$

Step 4. Simplify the expression by dividing out the common factors of $3$ , $x$ , $x$ , and $(x - 5)$ : $R(x) = \frac{x - 2}{3(x - 4)}$

Updated 2026-04-30

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OpenStax Intermediate Algebra

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