Mathematics » Rational Expressions and Equations » Add and Subtract Rational Expressions with Unlike Denominators

Finding the Least Common Denominator of Rational Expressions

Finding the Least Common Denominator of Rational Expressions

When we add or subtract rational expressions with unlike denominators we will need to get common denominators. If we review the procedure we used with numerical fractions, we will know what to do with rational expressions.

Let’s look at the example \(\cfrac{7}{12}+\cfrac{5}{18}\) from the tutorial on fractions. Since the denominators are not the same, the first step was to find the least common denominator (LCD). Remember, the LCD is the least common multiple of the denominators. It is the smallest number we can use as a common denominator.

To find the LCD of 12 and 18, we factored each number into primes, lining up any common primes in columns. Then we “brought down” one prime from each column. Finally, we multiplied the factors to find the LCD.

\(\cfrac{\begin{array}{} 12=2·2·3\hfill \\ 18=2·3·3\hfill \end{array}}{\begin{array}{}\\ \phantom{\rule{0.15em}{0ex}}\text{LCD}=2·2·3·3\hfill \\ \phantom{\rule{0.15em}{0ex}}\text{LCD}=36\hfill \end{array}}\)

We do the same thing for rational expressions. However, we leave the LCD in factored form.

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.

Remember, we always exclude values that would make the denominator zero. What values of x should we exclude in this next example?

Example

Find the LCD for \(\cfrac{8}{{x}^{2}-2x-3},\cfrac{3x}{{x}^{2}+4x+3}\).

Solution

\(\begin{array}{cccc}& & & \hfill \text{Find the LCD for}\phantom{\rule{0.2em}{0ex}}\cfrac{8}{{x}^{2}-2x-3},\cfrac{3x}{{x}^{2}+4x+3}.\hfill \\ \begin{array}{c}\text{Factor each expression completely, lining}\hfill \\ \text{up common factors.}\hfill \\ \text{Bring down the columns.}\hfill \\ \end{array}\hfill & & & \hfill \cfrac{\begin{array}{c}{x}^{2}-2x-3=\left(x+1\right)\left(x-2\right)\hfill \\ {x}^{2}+4x+3=\left(x+1\right)\phantom{\rule{1em}{0ex}}\left(x+3\right)\hfill \end{array}}{\phantom{\rule{4.45em}{0ex}}\text{LCD}=\left(x+1\right)\left(x-2\right)\left(x+3\right)}\hfill \\ \text{Multiply the factors.}\hfill & & & \hfill \text{The LCD is}\phantom{\rule{0.2em}{0ex}}\left(x+1\right)\left(x-3\right)\left(x+3\right).\hfill \end{array}\)

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