Simplifying Rational Expressions

Description/Explanation/Highlights

Video Description

This video explains how to simplify rational expressions using factoring and cancelling.

Steps and Key Points to Remember

To simplify a rational expression using factoring and cancelling, follow these steps:

  1. Rational expressions are fractional expressions with a variable in the denominator.
  2. To simplify a rational expression, factor the numerator and the denominator fully and look for common factors that will cancel.
  3. In the example, \(\frac{\textstyle14}{\textstyle21x+7}\), the denominator has a common factor of 7 that can be factored out as: \(\frac{\textstyle14}{\textstyle7(3x+1)}\). The number 14 in the numerator can be factored as \(7\cdot2\) giving us: \(\frac{\textstyle7(2)}{\textstyle7(3x+1)}\).
  4. The 7s cancel leaving the result as: \(\frac{\textstyle2}{\textstyle3x+1}\)
  5. In the example \(\frac{\textstyle2x+5}{\textstyle4x^2-25}\), the numerator does not factor but the denominator \(4x^2-25\) factors as the difference of two perfect squares into \((2x+5)(2x-5)\). The new problem looks like this: \(\frac{(\textstyle2x+5)}{\textstyle(2x+5)(2x-5)}\).
  6. There is now a common factor of \(2x+5\) in both the numerator and the denominator that will cancel leaving only a 1 in the numerator and the final answer of: \(\frac{\textstyle1}{\textstyle2x-5}\).
  7. Sometimes both the numerator and denominator can be factored as in the problem \(\frac{\textstyle x^2+2x-8}{\textstyle x^2+4x-12}\).
  8. The numerator is a trinomial and can be factored as \((x-2)(x+4)\). The denominator is also a trinomial and is factored as \((x-2)(x+6)\).
  9. Notice that the numerator and denominator have a common factor of \((x-2)\) which can be cancelled leaving the final simplified answer as \(\frac{\textstyle x+4}{\textstyle x+6}\)

Here are some key points to keep in mind when simplifying rational expression using factoring.

  • Begin the process of simplifying by factoring the numerator and denominator completely.
  • Sometimes factoring may be as simple as factoring out a common factor or even dividing constants into factors such as writing the constant 14 as 7(2) to match a constant that will cancel.
  • Cancel all common factors in the numerator and denominator to get the simplified rational expression.
  • Remember that if all factors in either the numerator or denominator are cancelled, the numerator or denominator becomes 1.

Video Highlights

  • 00:00 Introduction
  • 00:05 \(\frac{\textstyle14}{\textstyle21x+7}\) example of simplifying a rational expression
  • 01:25 \(\frac{\textstyle2x+5}{\textstyle4x^2-25}\) example of simplifying a rational expression with a difference of squares
  • 03:18 \(\frac{\textstyle x^2+2x-8}{\textstyle x^2+4x-12}\) example of simplifying a rational expression with two trinomials.
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