8.4 Add and subtract rational expressions with unlike denominators  (Page 3/4)

Subtract: $\frac{8y}{y^2-16}-\frac{4}{y-4}.$

Solution

Do the expressions have a common denominator? No. Rewrite each expression with the LCD.
Find the LCD.
Rewrite each rational expression as an equivalent rational expression with the LCD.
Simplify the numerators.
Subtract the rational expressions.
Simplify the numerators.
Factor the numerator to look for common factors.
Remove common factors.
Simplify.

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There are lots of negative signs in the next example. Be extra careful!

Subtract: $\frac{\mathrm{-3}n-9}{n^2+n-6}-\frac{n+3}{2-n}.$

Solution

Factor the denominator.
Since $n-2$ and $2-n$ are opposites, we will mutliply the second rational expression by $\frac{\mathrm{-1}}{\mathrm{-1}}$ .
Simplify.
Do the expressions have a common denominator? No.
Find the LCD.
Rewrite each rational expression as an equivalent rational expression with the LCD.
Simplify the numerators.
Simplify the rational expressions.
Somplify the numerator.
Factor the numerator to look for common factors.
Simplify.

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When one expression is not in fraction form, we can write it as a fraction with denominator 1.

Subtract: $\frac{5c+4}{c-2}-3.$

Solution

Write $3$ as $\frac{3}{1}$ to have 2 rational expressions.
Do the rational expressions have a common denominator? No.
Find the LCD of $c-2$ and $1.$ LCD = $c-2.$
Rewrite $\frac{3}{1}$ as an equivalent rational expression with the LCD.
Simplify.
Subtract the rational expressions.
Simplify.
Factor to check for common factors.
There are no common factors; the rational expression is simplified.

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We follow the same steps as before to find the LCD when we have more than two rational expressions. In the next example we will start by factoring all three denominators to find their LCD.

Simplify: $\frac{2u}{u-1}+\frac{1}{u}-\frac{2u-1}{u^2-u}.$

Solution

Do the rational expressions have a common denominator? No.
Find the LCD.
Rewrite each rational expression as an equivalent rational expression with the LCD.
Write as one rational expression.
Simplify.
Factor the numerator, and remove common factors.
Simplify.

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Key concepts

• Find the Least Common Denominator of Rational Expressions
  1. Factor each expression completely.
  2. List the factors of each expression. Match factors vertically when possible.
  3. Bring down the columns.
  4. Multiply the factors.
• Add or Subtract Rational Expressions
  1. Determine if the expressions have a common denominator.
     Yes – go to step 2.
     No – Rewrite each rational expression with the LCD.
     • Find the LCD.
     • Rewrite each rational expression as an equivalent rational expression with the LCD.
  2. Add or subtract the rational expressions.
  3. Simplify, if possible.

Practice makes perfect

In the following exercises, find the LCD.