Lesson 8.4: Add and Subtract Rational Expressions with Unlike Denominators

New Concept

To add or subtract rational expressions with different denominators, first find the Least Common Denominator (LCD). This lets you rewrite them as equivalent fractions, preparing you to combine the numerators and simplify the final result.

What’s next

Next, we'll break down finding the Least Common Denominator (LCD). You'll then apply this skill through interactive examples and practice cards.

Property

To add or subtract rational expressions with unlike denominators, we must first find the least common denominator (LCD). The LCD is the least common multiple of the denominators. We leave the LCD in factored form.

How to 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.

Examples

• Find the LCD for $\frac{5}{x^2 - 4}$ and $\frac{3x}{x^2 - x - 6}$. First, factor the denominators: $x^2 - 4 = (x-2)(x+2)$ and $x^2 - x - 6 = (x-3)(x+2)$. The LCD is the product of all unique factors: $(x-2)(x+2)(x-3)$.

Property

When we add or subtract fractions, once we find the LCD, we rewrite each fraction as an equivalent fraction with the LCD. To do this for rational expressions, we multiply the numerator and denominator of each expression by whatever factors are 'missing' from its denominator to make it match the LCD.

Examples

• Rewrite $\frac{7}{x+3}$ as an equivalent expression with the denominator $(x+3)(x-5)$. Multiply the numerator and denominator by the missing factor $(x-5)$: $\frac{7(x-5)}{(x+3)(x-5)} = \frac{7x-35}{(x+3)(x-5)}$.

• Rewrite $\frac{4x}{x^2-1}$ and $\frac{2}{x+1}$ with their LCD. The LCD is $(x-1)(x+1)$. The first fraction is already set. For the second, multiply by $(x-1)$: $\frac{2(x-1)}{(x+1)(x-1)} = \frac{2x-2}{x^2-1}$.

Property

How to add 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 the numerators and place the sum over the common denominator.
3. Simplify, if possible.

Examples

• Add $\frac{4}{x-3} + \frac{2}{x+5}$. The LCD is $(x-3)(x+5)$. This becomes $\frac{4(x+5)}{(x-3)(x+5)} + \frac{2(x-3)}{(x-3)(x+5)} = \frac{4x+20+2x-6}{(x-3)(x+5)} = \frac{6x+14}{(x-3)(x+5)}$.

• Add $\frac{3a}{a^2-9} + \frac{5}{a-3}$. The LCD is $(a-3)(a+3)$. This gives $\frac{3a}{(a-3)(a+3)} + \frac{5(a+3)}{(a-3)(a+3)} = \frac{3a+5a+15}{(a-3)(a+3)} = \frac{8a+15}{a^2-9}$.

Property

How to subtract rational expressions.

1. Determine if they 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. Subtract the numerators and place the difference over the common denominator.
3. Simplify, if possible.

Examples

• Subtract $\frac{x}{x+4} - \frac{x-1}{x-2}$. The LCD is $(x+4)(x-2)$. This gives $\frac{x(x-2)}{(x+4)(x-2)} - \frac{(x-1)(x+4)}{(x+4)(x-2)} = \frac{x^2-2x - (x^2+3x-4)}{(x+4)(x-2)} = \frac{-5x+4}{(x+4)(x-2)}$.

• Subtract $\frac{5b}{b^2-16} - \frac{2}{b+4}$. The LCD is $(b-4)(b+4)$. This becomes $\frac{5b}{(b-4)(b+4)} - \frac{2(b-4)}{(b-4)(b+4)} = \frac{5b - (2b-8)}{(b-4)(b+4)} = \frac{3b+8}{b^2-16}$.