8.4 Add and Subtract Rational Expressions with Unlike Denominators

Adding and Subtracting Rational Expressions with Unlike Denominators

Adding and subtracting rational expressions with unlike denominators works just like adding numeric fractions: you need a common denominator before you can combine anything. The difference is that with algebraic expressions, finding that common denominator often requires factoring polynomials.

Least Common Denominator for Rational Expressions

The least common denominator (LCD) is the smallest expression that each denominator divides into evenly. Here's how to find it:

1. Factor each denominator completely (into primes for numbers, or into polynomial factors for algebraic expressions).
2. List every distinct factor that appears in any denominator.
3. For each factor, take the highest power that appears.
4. Multiply those together. That product is your LCD.

Numeric example: Find the LCD of $\frac{2}{15}$ and $\frac{3}{20}$

• $15 = 3 \times 5$
• $20 = 2^2 \times 5$
• Highest power of 2: $2^2$. Highest power of 3: $3^1$. Highest power of 5: $5^1$.
• LCD = $2^2 \times 3 \times 5 = 60$

Algebraic example: Find the LCD of $\frac{5}{x^2 - 4}$ and $\frac{3}{x + 2}$

• $x^2 - 4 = (x+2)(x-2)$
• $x + 2$ is already fully factored
• LCD = $(x+2)(x-2)$, since that already contains $x+2$

Conversion to Common Denominators

Once you have the LCD, you convert each fraction so its denominator equals the LCD. You do this by multiplying the numerator and denominator by the same factor, which keeps the expression equivalent.

That factor is whatever's "missing" from the current denominator compared to the LCD. In other words, divide the LCD by the current denominator.

Example: Convert $\frac{2}{15}$ and $\frac{3}{20}$ to equivalent forms with the LCD of 60.

• For $\frac{2}{15}$: the missing factor is $\frac{60}{15} = 4$, so $\frac{2}{15} \times \frac{4}{4} = \frac{8}{60}$
• For $\frac{3}{20}$: the missing factor is $\frac{60}{20} = 3$, so $\frac{3}{20} \times \frac{3}{3} = \frac{9}{60}$

The same idea applies to algebraic fractions. If the LCD is $(x+2)(x-2)$ and your denominator is $(x+2)$, you multiply top and bottom by $(x-2)$.

Addition of Unlike Rational Expressions

1. Find the LCD of all denominators.
2. Convert each fraction to an equivalent form with the LCD.
3. Add the numerators. Keep the common denominator.
4. Simplify the result by factoring and canceling common factors.

Example: $\frac{2}{15} + \frac{3}{20}$

1. LCD = 60
2. $\frac{8}{60} + \frac{9}{60}$
3. $\frac{8 + 9}{60} = \frac{17}{60}$
4. 17 is prime and doesn't divide 60, so $\frac{17}{60}$ is already simplified.

Subtraction of Unlike Rational Expressions

The process is identical to addition, except you subtract the numerators in step 3. Be careful with signs: distributing the negative across the entire second numerator is where most mistakes happen.

Example: $\frac{7}{12} - \frac{5}{18}$

1. Factor: $12 = 2^2 \times 3$ and $18 = 2 \times 3^2$. LCD = $2^2 \times 3^2 = 36$.

2. $\frac{7}{12} \times \frac{3}{3} = \frac{21}{36}$ and $\frac{5}{18} \times \frac{2}{2} = \frac{10}{36}$

3. $\frac{21 - 10}{36} = \frac{11}{36}$

4. 11 is prime and doesn't divide 36, so the answer is $\frac{11}{36}$.

Working with Algebraic Fractions

When variables appear in the denominators, the same four-step process applies. The main added challenge is that you'll need to factor polynomials to find the LCD.

Example: $\frac{3}{x+1} + \frac{5}{x-2}$

1. The denominators $(x+1)$ and $(x-2)$ share no common factors, so the LCD = $(x+1)(x-2)$.

2. Multiply each fraction by its missing factor:

   • $\frac{3}{x+1} \times \frac{x-2}{x-2} = \frac{3(x-2)}{(x+1)(x-2)}$
   • $\frac{5}{x-2} \times \frac{x+1}{x+1} = \frac{5(x+1)}{(x+1)(x-2)}$

3. Add the numerators: $\frac{3(x-2) + 5(x+1)}{(x+1)(x-2)} = \frac{3x - 6 + 5x + 5}{(x+1)(x-2)} = \frac{8x - 1}{(x+1)(x-2)}$

4. The numerator $8x - 1$ doesn't factor further and shares no common factors with the denominator, so this is the final answer.

When the denominators contain factorable expressions (like $x^2 - 9 = (x+3)(x-3)$), always factor first before determining the LCD. Skipping the factoring step often leads to an LCD that's larger than necessary, which makes the arithmetic harder and the simplification messier.