Multiplying Rational Expressions- Methods and Examples
What Are Rational Expressions?
A rational expression is simply a fraction where both the top and bottom are polynomials. That's it. Nothing fancy. When you multiply two rational expressions together, you're doing the same thing as multiplying two fractions—just with more complicated numbers involved.
The good news: if you know how to multiply fractions, you already know how to do this. The tricky part is the factoring and simplifying you'll need to do along the way.
Before You Start: The Prerequisite Skills
You need to be solid on these before attempting to multiply rational expressions:
- Factoring polynomials (especially trinomials and difference of squares)
- Canceling common factors
- Multiplying fractions
- Identifying when a fraction can be simplified
If you're shaky on factoring, stop here. Go practice that first. Trying to multiply rational expressions without factoring skills is like trying to drive without knowing how to steer.
The Basic Method
Here's the straightforward approach:
- Multiply all numerators together
- Multiply all denominators together
- Factor everything you can
- Cancel any common factors between numerators and denominators
- Write your final answer
Seems simple. The catch is step 3—factoring correctly. That's where most people mess up.
Example 1: Straight Multiplication
Multiply: (x + 2) / 3 · 5 / (x - 1)
Step 1: Multiply across
(x + 2) · 5 / 3 · (x - 1) = 5(x + 2) / 3(x - 1)
Step 2: Check for cancellation
There's nothing that cancels here. The numerator has 5(x + 2) and the denominator has 3(x - 1). No common factors.
Final answer: 5(x + 2) / 3(x - 1)
Example 2: Multiplication with Factoring Required
Multiply: (x² - 9) / (x² + 5x + 6) · (x + 2) / (x - 3)
Step 1: Factor everything
x² - 9 = (x + 3)(x - 3) [difference of squares]
x² + 5x + 6 = (x + 2)(x + 3) [factoring the trinomial]
Now the problem looks like:
[(x + 3)(x - 3)] / [(x + 2)(x + 3)] · (x + 2) / (x - 3)
Step 2: Cancel common factors
- (x + 3) in numerator cancels with (x + 3) in denominator
- (x + 2) in numerator cancels with (x + 2) in denominator
- (x - 3) in numerator cancels with (x - 3) in denominator
Step 3: What's left?
Nothing. Everything canceled.
Final answer: 1
Example 3: More Complex
Multiply: (2x² + 8x) / (x² - 4) · (x² - 5x + 6) / (4x²)
Step 1: Factor everything
2x² + 8x = 2x(x + 4)
x² - 4 = (x + 2)(x - 2)
x² - 5x + 6 = (x - 2)(x - 3)
4x² = 4x · x
The expression becomes:
[2x(x + 4)] / [(x + 2)(x - 2)] · [(x - 2)(x - 3)] / [4x · x]
Step 2: Cancel
- 2x in numerator with 4x · x in denominator → leaves 1 / (2x)
- (x - 2) cancels
- (x + 4) stays
- (x + 2) stays
- (x - 3) stays
Step 3: Multiply what remains
(x + 4)(x - 3) / 2x(x + 2)
Final answer: (x + 4)(x - 3) / 2x(x + 2)
The Smart Approach: Cancel BEFORE You Multiply
Most textbooks tell you to multiply first, then simplify. That's inefficient. The better method: cancel factors before multiplying.
Look at the factored form of your problem. Cancel anything that appears in both a numerator and a denominator. Then multiply what's left. You'll deal with smaller numbers and avoid messy intermediate steps.
Common Mistakes That Will Cost You Points
- Forgetting to factor — you cannot cancel terms that aren't factored. You can only cancel factors, not terms.
- Canceling across addition — you can't cancel the x in "x + 5" with something else. Only cancel factors that are multiplied, not added.
- Dropping denominators — if a denominator becomes zero after substituting a value, that value is excluded from the domain. Note your restrictions.
- Not checking for restrictions — any value that makes a denominator zero is not allowed. Write these down.
Domain Restrictions: The Part Everyone Forgets
Every rational expression has restrictions based on what makes denominators equal zero.
In the original problem: (x² - 9) / (x² + 5x + 6) · (x + 2) / (x - 3)
The denominators are (x² + 5x + 6) and (x - 3). Factor these:
x² + 5x + 6 = (x + 2)(x + 3)
x - 3 = (x - 3)
Restrictions: x ≠ -2, x ≠ -3, x ≠ 3
Even if everything cancels to 1, these values are still excluded. The simplified answer doesn't change the domain.
Quick Reference: Factoring Patterns to Memorize
| Pattern | Factored Form |
|---|---|
| Difference of squares | a² - b² = (a + b)(a - b) |
| Perfect square trinomial | a² + 2ab + b² = (a + b)² |
| Perfect square trinomial | a² - 2ab + b² = (a - b)² |
| Common trinomial | x² + bx + c = (x + m)(x + n) where m+n = b and mn = c |
Getting Started: Your Action Plan
When you see a multiplication problem with rational expressions:
- Factor every polynomial in every numerator and denominator
- Cross out matching factors between any numerator and any denominator
- Multiply what's left in the numerators together
- Multiply what's left in the denominators together
- List your restrictions from the original denominators
Practice with 10 problems using this method. By the fifth one, it'll feel automatic. The factoring is what slows you down—get faster at that and these problems become trivial.
When to Simplify During the Process
Simplify whenever you see common factors. Don't wait until the end. Cancel as soon as you spot them. This keeps your numbers manageable and reduces errors.
If nothing cancels and everything is already factored, you're done. Don't force simplification that isn't there.