Example

Dividing the Complex Fraction 6x2−7x+24x−82x2−7x+3x2−5x+6\frac{\frac{6x^2-7x+2}{4x-8}}{\frac{2x^2-7x+3}{x^2-5x+6}}

Divide 6x2−7x+24x−82x2−7x+3x2−5x+6\frac{\frac{6x^2-7x+2}{4x-8}}{\frac{2x^2-7x+3}{x^2-5x+6}}.

Because a fraction bar represents division, a complex fraction is simply another way of writing the division of two fractions.

Step 1 — Rewrite with a division sign. The main fraction bar separates the numerator fraction from the denominator fraction:

6x2−7x+24x−8÷2x2−7x+3x2−5x+6\frac{6x^2-7x+2}{4x-8} \div \frac{2x^2-7x+3}{x^2-5x+6}

Step 2 — Rewrite as the product of the first fraction and the reciprocal of the second:

6x2−7x+24x−8⋅x2−5x+62x2−7x+3\frac{6x^2-7x+2}{4x-8} \cdot \frac{x^2-5x+6}{2x^2-7x+3}

Step 3 — Factor the numerators and denominators, then multiply. Factor each polynomial:

  • 6x2−7x+2=(2x−1)(3x−2)6x^2-7x+2 = (2x-1)(3x-2)
  • 4x−8=4(x−2)4x-8 = 4(x-2)
  • x2−5x+6=(x−2)(x−3)x^2-5x+6 = (x-2)(x-3)
  • 2x2−7x+3=(2x−1)(x−3)2x^2-7x+3 = (2x-1)(x-3)

The expression becomes:

(2x−1)(3x−2)(x−2)(x−3)4(x−2)(2x−1)(x−3)\frac{(2x-1)(3x-2)(x-2)(x-3)}{4(x-2)(2x-1)(x-3)}

Step 4 — Simplify by dividing out common factors. Cancel the shared factors (2x−1)(2x-1), (x−2)(x-2), and (x−3)(x-3):

3x−24\frac{3x-2}{4}

This example shows that when a division of rational expressions is written as a complex fraction (one fraction stacked over another), the first step is to rewrite the main fraction bar as a division sign. From there, the standard procedure applies: multiply by the reciprocal, factor, and cancel.

0

1

Updated 2026-04-21

Contributors are:

Who are from:

Tags

OpenStax

Elementary Algebra @ OpenStax

Ch.8 Rational Expressions and Equations - Elementary Algebra @ OpenStax

Algebra

Math

Prealgebra

Related
Learn After