Multiplying Rational Expressions

Description/Explanation/Highlights

Video Description

This video explains how to multiply rational expressions.

Steps and Key Points to Remember

To multiply rational expressions, follow these steps:

  1. Just as with addition or subtraction of rational expressions, the rules of fractions apply to multiplication of rational expressions.
  2. When multiplying two fractions together, we look to reduce or cancel using common factors and then multiply the numerators together and the denominators together. We will do the same thing to multiply rational expressions.
  3. In the example, \(\frac{\textstyle x^2-9}{\textstyle x^2-2x}\cdot\frac{\textstyle 2x-4}{\textstyle x^2-x-12}\), look at each numerator and denominator separately and determine if it can be factored.
  4. \(x^2-9\)  factors into \((x+3)(x-3)\), an \(x\) can be factored out of \(x^2-2x\) leaving \(x(x-2)\), a 2 can be factored out of \(2x-4\) leaving \(2(x-2)\), and finally \(x^2-x-12\) factors to \((x+3)(x-4)\).
  5. Write the problem out in factored form: \(\frac{\textstyle (x+3)(x-3)}{\textstyle x(x-2)}\cdot\frac{\textstyle 2(x-2)}{\textstyle (x+3)(x-4)}\)
  6. Now, look for factors in any numerator that are the same as factors in any denominator. These are equal to 1 and will effectively cancel.
  7. The \(x+3\) in the first numerator will cancel the \(x+3\) in the second denominator, and the \(x-2\) in the second numerator will cancel the \(x-2\) in the first denominator leaving: \(\frac{\textstyle 2(x-3)}{\textstyle x(x-4)}\).
  8. Complete the problem by distributing in the numerator and denominator to get the final answer. \(\frac{\textstyle 2x-6}{\textstyle x^2-4x}\)

Here are some key points to keep in mind when multiplying rational expressions.

  • Follow the rules that you would use to multiply any fraction to multiply rational expressions.
  • Factor completely all of the numerators and denominators first.
  • Look for common factors that will cancel from numerators and denominators.
  • After cancelling multiply the numerators together and the denominators together by distributing and simplifying as needed.
  • Factors can be cancelled from any numerator with factors from any denominator in the problem.

Video Highlights

  • 00:00 Introduction
  • 00:07 \(\frac{\textstyle x^2-9}{\textstyle x^2-2x}\cdot\frac{\textstyle 2x-4}{\textstyle x^2-x-12}\) example and steps of multiplying rational expressions.
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