1Cademy - Example of Dividing the Rational Functions $$f(x) = \frac{15x^2}{3x^2 + 33x}$$ and $$g(x) = \frac{5x - 5}{x^2 + 9x

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How to Divide Rational Expressions

Example

Example of Dividing the Rational Functions $f(x) = \frac{15x^2}{3x^2 + 33x}$ and $g(x) = \frac{5x - 5}{x^2 + 9x - 22}$

To find the quotient $R(x) = \frac{f(x)}{g(x)}$ for the rational functions $f(x) = \frac{15x^2}{3x^2 + 33x}$ and $g(x) = \frac{5x - 5}{x^2 + 9x - 22}$ , follow the procedure for dividing rational expressions:

Step 1. Substitute the given polynomial fractions into the division operation: $R(x) = \frac{\frac{15x^2}{3x^2 + 33x}}{\frac{5x - 5}{x^2 + 9x - 22}}$

Step 2. Rewrite the division as the product of $f(x)$ and the reciprocal of $g(x)$ : $R(x) = \frac{15x^2}{3x^2 + 33x} \cdot \frac{x^2 + 9x - 22}{5x - 5}$

Step 3. Factor each numerator and denominator completely, then multiply: $R(x) = \frac{15x^2(x + 11)(x - 2)}{3x(x + 11) \cdot 5(x - 1)}$

Expanding the numerical and variable factors to identify commonalities: $R(x) = \frac{3 \cdot 5 \cdot x \cdot x \cdot (x + 11)(x - 2)}{3 \cdot x \cdot (x + 11) \cdot 5 \cdot (x - 1)}$

Step 4. Simplify by dividing out the common factors of $3$ , $5$ , $x$ , and $(x + 11)$ : $R(x) = \frac{x(x - 2)}{x - 1}$

Updated 2026-04-30

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OpenStax Intermediate Algebra

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