Math 101 Intermediate Algebra
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Addition and Subtraction of Rational Expressions
Chapter 7, Section 2


Learning Plan

  1. Learn how to add and subtract rational expressions already having a common denominator.

    • A common denominator is necessary to add and subtract any rational expressions.

  2. Learn how to find the least common denominator.

  3. Learn how to add and subtract rational expressions having unlike denominators.


Adding and Subtracting Rational Expressions having a Common Denominator

Given any expressions a, b, and c with c 0:

Addition Subtraction
a
c
  +   b
c
  =   a + b
c
a
c
  -   b
c
  =   a - b
c

To add (or subtract) rational expressions already having a common denominator.....

  1. Add (or subtract) the numerators.

  2. Place the sum (or difference) of the numerators found in step 1 over the common denominator.

  3. If possible, factor the resulting numerator and reduce fractions.


Finding the Least Common Denominator

To find the least common denominator.....

  1. Factor each denominator completely.
    • Any factors that appear more than once in one denominator should be expressed as powers.
    • For example, (x + 5)(x + 5) should be written (x + 5)2.

  2. List all different factors that appear in any of the denominators.
    • When the same factor appears in more than one denominator, write the factor with the highest power that appears.

  3. The least common denominator is the product of all the factors found in step 2.


Adding and Subtracting Rational Expressions having Unlike Denominators

To add (or subtract) rational expressions having unlike denominators.....

  1. Determine the least common denominator.

  2. Rewrite each fraction as an equivalent fraction having the least common denominator.
    • This is done by multiplying both the numerator and denominator of each fraction by any factors needed to obtain the least common denominator.

  3. Multiply out the numerators while leaving the denominators in factored form.
    • You now have rational expressions having a common denominator.

  4. Add (or subtract) the rational expressions as was done above in the "adding (or subtracting) rational expressions already having a common denominator" section.