Subtracting Rational Expressions with Different Denominators

Description/Explanation/Highlights

Video Description

This video explains how to subtract rational expressions when the denominators are different.

Steps and Key Points to Remember

To subtract rational expressions with different denominators, follow these steps:

  1. Just as with addition, in order to subtract rational expressions, the denominators must be the same.
  2. When the denominators are not the same, we must find a common denominator before solving the rest of the problem.
  3. To find a common denominator, start by simplifying all the denominators by factoring if possible.
  4. Once the denominators are simplified, find the least common denominator (LCD) by listing all the unique terms in all the denominators. This will become our common denominator.
  5. Once the LCD has been determined, the numerators of each expression should be multiplied by any terms in the LCD that were not in the original denominator.
  6. Place the new numerators over the common denominator and combine terms, taking care to distribute the minus sign to all the terms in the numerator of the second expression as you would if the expressions had common denominators. (If you don’t know how to do this refer to the video: Adding and Subtracting Rational Expressions with a Common Denominator)
  7. In the example \(\frac{\textstyle x}{\textstyle x^2+2x-3}-\frac{\textstyle 2x+2}{\textstyle 4x+12}\), the denominators are different so we must first find a common denominator and adjust the numerators.
  8. The first step is to simplify the denominators as much as possible. In this case, factor a 4 out of the second denominator and factor the trinomial in the first denominator completely, rewriting as: \(\frac{\textstyle x}{\textstyle (x+3)(x-1)}-\frac{\textstyle 2x+2}{\textstyle 4(x+3)}\).
  9. The next step is to find the LCD by listing all the unique factors in the denominators. List all the factors in the first denominator which are \((x+3)\) and \((x-1)\). The \((x+3)\) in the second denominator is not unique since it was already used in the first but the \(4\) is unique so it must be added as part of the LCD.
  10. The LCD in this example is \(4(x+3)(x-1)\).
  11. Next, multiply each numerator by anything in the LCD that was not in the original denominator for that expression.
  12. In this example \(4\) was not in the denominator of the first expression so the numerator of the first expression must be multiplied by \(4\).
  13. \((x-1)\) is not in the second denominator so the second numerator is multiplied by \((x-1)\).
  14. Write all this over the LCD which is the new common denominator as follows: \(\frac{\textstyle 4x-(2x+2)(x-1)}{\textstyle 4(x+3)(x-1)}\)
  15. Now distribute and combine like terms in the numerator and denominator.  Put a parenthesis around the numerator of the second expression until the negative sign can be distributed. \(\frac{\textstyle 4x-(2x^2-2x+2x-2)}{\textstyle (4x+12)(x-1)}\)
  16. Next combine the \(-2x \) and \(2x\) in the numerator to get rid of the x and then distribute the negative sign. Multiply the binomials in the denominator. \(\frac{\textstyle 4x-2x^2+2}{\textstyle  4x^2-4x+12x-12}\).
  17. Finally, combine and put in order: \(\frac{\textstyle -2x^2+4x+2}{\textstyle 4x^2+8x-12}\). This is the final answer.

Here are some key points to keep in mind when subtracting rational expressions with different denominators.

  • The denominators must be the same (common) before subtracting numerators.
  • Find a common denominator by finding a least common denominator (LCD).
  • Before finding the LCD, first simplify the denominators by factoring.
  • The LCD is made up of all the unique terms in all the denominators.
  • Don’t put the same term in the LCD more than once unless it is used in one of the original denominators more than once.
  • Multiply numerators by terms in the LCD that are “missing” in the original denominator.
  • Put the new numerators over the common denominator and simplify both to get the final answer.
  • It is a good idea to use parenthesis around the second numerator until the negative sign has been distributed.
  • Be careful with the negative sign. It must be distributed to all terms in the second numerator.

Video Highlights

  • 00:00 Introduction
  • 00:05 \(\frac{\textstyle x}{\textstyle x^2+2x-3}-\frac{\textstyle 2x+2}{\textstyle 4x+12}\) example of subtracting rational expressions with different denominators.
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