Adding and Subtracting Rational Expressions with a Common Denominator

Description/Explanation/Highlights

Video Description

This video explains how to add and subtract rational expressions when there is a common denominator.

Steps and Key Points to Remember

To add or subtract rational expressions that have a common denominator, follow these steps:

  1. Operations on rational expressions follow the same rules as operations on any fraction.
  2. When there is a common denominator, combine the numerators and keep the denominators.
  3. In the example \(\frac{\textstyle 2x+5}{\textstyle x-7}+\frac{\textstyle 3x-2}{\textstyle x-7}\), the common denominator is \(x-7\) so combine the numerators by combining like terms and put the result over the common denominator.
  4. Combining \(2x+3x=5x\) and \(5-2=3\) so the numerator is \(5x+3\) and after putting it over the common denominator the final answer is \(\textstyle\frac{\textstyle 5x+3}{\textstyle x-7}\)
  5.  Subtraction is done the same way but care must be taken to distribute the negative sign to all terms in the expression being subtracted.
  6. In the example \(\frac{\textstyle 4x^2-5x}{\textstyle 2x^2-1}-\frac{\textstyle 3x^2+2x-1}{\textstyle 2x^2-1}\), there is a common denominator of \(2x^2-1\) so combine the numerators taking care to distribute the minus sign to all terms in the second expression.
  7. It may be helpful to write the numerators over the common denominator adding parenthesis as follows: \(\frac{\textstyle 4x^2-5x-(3x^2+2x-1)}{\textstyle 2x^2-1}\).
  8. Simplify the numerator by distributing the negative to each term in the parenthesis (changing each sign) \(\frac{\textstyle 4x^2-5x-3x^2-2x+1}{\textstyle 2x^2-1}\) and then combining like terms in the numerator to get the final answer \(\frac{\textstyle x^2-7x+1}{\textstyle 2x^2-1}\).

Here are some key points to keep in mind when adding and subtracting rational expressions that have common denominators.

  • Follow the same rules as addition or subtraction of any fraction.
  • The denominators must be the same (common).
  • Combine like terms in the numerator and put the result over the common denominator.
  • If subtracting, pay careful attention to the signs, making sure that every sign in the expression being subtracted is changed.
  • It may be helpful to write the entire problem over the common denominator and add parenthesis when subtracting before combining like terms.

Video Highlights

  • 00:00 Introduction
  • 00:08 \(\frac{\textstyle 2x+5}{\textstyle x-7}+\frac{\textstyle 3x-2}{\textstyle x-7}\) example of adding rational expressions that have a common denominator.
  • 00:54 \(\frac{\textstyle 4x^2-5x}{\textstyle 2x^2-1}-\frac{\textstyle 3x^2+2x-1}{\textstyle 2x^2-1}\) example of subtracting rational expressions that have a common denominator.
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