Lesson 95: Combining Rational Expressions with Unlike Denominators

New Concept

A least common denominator (LCD) is the least common multiple (LCM) of the denominators.

What’s next

Next, you’ll apply this principle to add and subtract rational expressions by finding the LCD and creating equivalent fractions.

Property

A least common denominator (LCD) is the least common multiple (LCM) of the denominators.

Explanation

Think of denominators as different languages; you can't add fractions until they speak the same one! To find the LCD, factor each denominator completely. Then, build the LCD by taking every unique factor the greatest number of times it appears in any single denominator. This process creates a new, shared foundation for your fractions.

Examples

For $\frac{5}{(x-2)}$ and $\frac{7}{(x-2)(x+3)}$, the LCD is $(x-2)(x+3)$.
For $\frac{2x}{3(x+1)}$ and $\frac{9}{x^2-1}$, factor to get $\frac{2x}{3(x+1)}$ and $\frac{9}{(x-1)(x+1)}$. The LCD is $3(x+1)(x-1)$.
For $\frac{a}{2(a-5)}$ and $\frac{b}{4(a-5)}$, the LCD is $4(a-5)$.

Property

To add or subtract rational expressions with unlike denominators, first find the LCD. Then, rewrite each expression as an equivalent fraction with the LCD. Finally, add or subtract the numerators and place the result over the common denominator.

Explanation

You can't just smash fractions together! First, find their Least Common Denominator (LCD). Give each fraction a makeover by multiplying its top and bottom by the missing pieces of the LCD. Once they have matching denominators, you can finally combine their numerators. It’s all about making sure every term is playing on the same field!

Examples

$\frac{3}{x+1} + \frac{5}{x-1} = \frac{3(x-1)}{(x+1)(x-1)} + \frac{5(x+1)}{(x+1)(x-1)} = \frac{8x+2}{(x+1)(x-1)}$
$\frac{x}{x-4} - \frac{2}{3(x-4)} = \frac{3x}{3(x-4)} - \frac{2}{3(x-4)} = \frac{3x-2}{3(x-4)}$
$\frac{2}{x^2-9} + \frac{1}{x+3} = \frac{2}{(x-3)(x+3)} + \frac{1(x-3)}{(x-3)(x+3)} = \frac{x-1}{x^2-9}$

Combining different algebraic fractions is like finding a common language. Let's see how with this key idea of adding rational expressions with unlike denominators.

Example Problem:
Add the expressions $\frac{5x^2}{x^2 - 9} + \frac{x+2}{3x-9}$.

Step-by-Step:

1. First, factor each denominator to identify the building blocks.

2. The least common denominator (LCD) must contain every factor from both denominators. The LCD is $3(x-3)(x+3)$.
3. Now, write an equivalent fraction for each term using the LCD.
   • For the first fraction, multiply the numerator and denominator by $3$:

• For the second fraction, multiply the numerator and denominator by $(x+3)$:

4. With common denominators, we can add the numerators.

5. Finally, expand and combine like terms in the numerator to simplify.

Property

Always check to see if the numerator of the sum can be factored and if a common factor can be divided out.

Explanation

Don't stop after adding or subtracting, because your final answer might be wearing a disguise! Always try to factor the final numerator. If you find a factor that matches one in the denominator, you can divide them out to simplify the expression. It’s like cleaning up your work to find the neatest possible answer.

Examples

$\frac{5x+10}{3(x+2)} = \frac{5(x+2)}{3(x+2)} = \frac{5}{3}$
$\frac{x^2-16}{x^2+x-20} = \frac{(x-4)(x+4)}{(x+5)(x-4)} = \frac{x+4}{x+5}$
$\frac{2x-8}{2x^2 - 32} = \frac{2(x-4)}{2(x^2-16)} = \frac{2(x-4)}{2(x-4)(x+4)} = \frac{1}{x+4}$

Subtraction can be tricky, but watch how simplifying at the end makes this problem elegant. This example demonstrates this key idea of subtracting rational expressions, including simplification.

Example Problem:
Subtract the expressions $\frac{x+1}{2x-10} - \frac{x+13}{6x-30}$.

Step-by-Step:

1. Begin by factoring each denominator to find their components.

2. Identify the least common denominator (LCD). Here, the LCD is $6(x-5)$.
3. Rewrite the first fraction as an equivalent fraction with the LCD. The second fraction already has the LCD's factors.

4. Combine the fractions by subtracting the numerators. Be careful to distribute the negative sign.

5. Expand the expression in the numerator and collect like terms.

6. Factor the numerator to check for common factors with the denominator.

7. Divide out the common factor $(x-5)$ to simplify the expression.