Dividing Rational Expressions

Description/Explanation/Highlights

Video Description

This video explains how to divide rational expressions.

Steps and Key Points to Remember

To divide rational expressions, follow these steps:

  1. Just as with addition, subtraction or multiplication of rational expressions, the operational rules of fractions apply to division of rational expressions.
  2. It is very useful to understand how rational expressions are multiplied since every division problem turns into a multiplication problem. If you need help multiplying rational expressions, please watch my video, Multiplying Rational Equations.
  3. To solve a division problem involving rational expressions, multiply the first expression by the reciprocal of the second. In other words, flip the denominator and numerator of the second expression and apply the rules of multiplying rational expressions.
  4. In the example, \(\frac{\textstyle x^2+5x+6}{\textstyle x^2-5x+4}\div\frac{\textstyle 3x^2+15x+18}{\textstyle x^2-2x+1}\), we must multiply by the reciprocal of the second expression so we should flip it and rewrite as a multiplication problem like: \(\frac{\textstyle x^2+5x+6}{\textstyle x^2-5x+4}\cdot\frac{\textstyle x^2-2x+1}{\textstyle 3x^2+15x+18}\)
  5. When multiplying two rational expressions together, we look to reduce or cancel using common factors and then multiply the numerators together and the denominators together.
  6. After flipping the second expression, factor each numerator and denominator completely.
  7. Rewrite the multiplication problem in factored form. 
  8. \(\frac{\textstyle (x+3)(x+2)}{\textstyle (x-4)(x-1)}\cdot\frac{\textstyle (x-1)(x-1)}{\textstyle 3(x+3)(x+2)}\)
  9. 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.
  10. Both factors in the numerator of the first expression will cancel with factors in the denominator of the second expression, and one of the \(x-1\) factors in the second numerator cancels with the first denominator. Note: We can only cancel one of the \(x-1\) factors since it appears only once in the denominator.
  11. The result after cancelling will be: \(\frac{\textstyle x-1}{\textstyle 3(x-4)}\)
  12. Distribute the 3 in the denominator for the final answer of \(\frac{\textstyle x-1}{\textstyle 3x-12}\)

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

  • All division problems become multiplication problems so understanding the rules of multiplication of rational expressions is essential. Click here for Multiplying Rational Expression Video.
  • To divide a rational expression, flip the second expression and change division to multiplication.
  • 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
  • 00:06 \(\frac{\textstyle x^2+5x+6}{\textstyle x^2-5x+4}\div\frac{\textstyle 3x^2+15x+18}{\textstyle x^2-2x+1}\) example of dividing rational expressions.
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  • To watch this video on YouTube in a new window with clickable highlights, click here

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Subtracting Rational Expressions with Different Denominators
Multiplying Rational Expressions
Solving Rational Equations
Solving More Complex Rational Equations
Translations of the Rational Parent Function
Reflections and Vertical Stretches of the Rational Parent Function
Vertical Compressions and Multiple Transformations of the Rational Parent
Finding Vertical Asymptotes
Finding Horizontal Asymptotes
Finding Removable Discontinuities
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