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Example of Dividing the Rational Functions f(x)=3x2x2−4xf(x) = \frac{3x^2}{x^2 - 4x} and g(x)=9x2−45xx2−7x+10g(x) = \frac{9x^2 - 45x}{x^2 - 7x + 10}

To find the quotient R(x)=f(x)g(x)R(x) = \frac{f(x)}{g(x)} for the rational functions f(x)=3x2x2−4xf(x) = \frac{3x^2}{x^2 - 4x} and g(x)=9x2−45xx2−7x+10g(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)=3x2x2−4x9x2−45xx2−7x+10R(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)f(x) and the reciprocal of g(x)g(x): R(x)=3x2x2−4x⋅x2−7x+109x2−45xR(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)=3⋅x⋅x⋅(x−5)(x−2)x(x−4)⋅3⋅3⋅x⋅(x−5)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, xx, xx, and (x−5)(x - 5): R(x)=x−23(x−4)R(x) = \frac{x - 2}{3(x - 4)}

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Updated 2026-07-01

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References

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OpenStax

Intermediate Algebra @ OpenStax

Ch.7 Rational Expressions and Functions - Intermediate Algebra @ OpenStax

Algebra

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