Multiply Rational Expressions- Step-by-Step Tutorial
What Are Rational Expressions?
A rational expression is simply a fraction where both the top and bottom are polynomials. That's it. Nothing fancy.
For example, (x + 2) over (x - 3) is a rational expression. So is (x² - 4) over (x + 2). You can add, subtract, multiply, and divide them — just like regular fractions.
This tutorial focuses only on multiplication. If you need division, convert it to multiplication by flipping the second fraction first.
The Core Rule for Multiplying Rational Expressions
Multiply the numerators together. Multiply the denominators together. Then simplify if possible.
In math terms:
(A/B) × (C/D) = (A × C) / (B × D)
That sounds easy. The catch? You almost never want to multiply everything out first. That's a rookie mistake that makes your life miserable.
The Right Way: Factor First, Cancel Second
Here's the process that actually works:
- Factor every numerator and denominator completely
- Cancel any factors that appear in both a numerator and denominator
- Multiply what remains
Skipping step one is why most people get stuck. You can't cancel terms that aren't factored. You can only cancel factors, not terms.
Why Factor First?
Let's say you have (x² - 9) over (x + 3). If you multiply everything out without factoring, you're stuck. But if you factor first:
(x² - 9) = (x + 3)(x - 3)
Now you can cancel the (x + 3) from top and bottom. That's the move.
Step-by-Step Example
Example 1: Basic Multiplication
Multiply: (x + 2)/(x - 1) × (x - 3)/(x + 4)
Step 1: Both fractions are already factored. No common factors between any numerator and denominator.
Step 2: Multiply the numerators: (x + 2)(x - 3)
Step 3: Multiply the denominators: (x - 1)(x + 4)
Step 4: Write the answer: (x + 2)(x - 3) / (x - 1)(x + 4)
You can leave it factored or multiply it out. Either works, but factored form makes it easier to spot if you missed any cancellations.
Example 2: Canceling Before Multiplying
Multiply: (x² - 4)/(x + 1) × (x + 2)/(x² - 1)
Step 1: Factor everything.
- x² - 4 = (x + 2)(x - 2)
- x² - 1 = (x + 1)(x - 1)
The expression becomes:
[(x + 2)(x - 2)]/(x + 1) × (x + 2)/[(x + 1)(x - 1)]
Step 2: Cancel common factors. You have (x + 2) in the first numerator and (x + 1) appearing in both denominators.
Cancel one (x + 2) from top with one from bottom.
Cancel one (x + 1) from top with one from bottom.
What's left?
(x - 2)/(1) × (1)/(x - 1)
Step 3: Multiply: (x - 2)/(x - 1)
That's your final answer. Much simpler than multiplying everything out first and then trying to factor again.
Example 3: Trinomial Factoring
Multiply: (x² + 5x + 6)/(x² + 4x + 3) × (x² + 2x - 3)/(x² + 6x + 8)
Step 1: Factor each polynomial.
- x² + 5x + 6 = (x + 2)(x + 3)
- x² + 4x + 3 = (x + 1)(x + 3)
- x² + 2x - 3 = (x + 3)(x - 1)
- x² + 6x + 8 = (x + 2)(x + 4)
The expression becomes:
[(x + 2)(x + 3)]/[(x + 1)(x + 3)] × [(x + 3)(x - 1)]/[(x + 2)(x + 4)]
Step 2: Cancel common factors.
- (x + 3) appears three times across numerators, cancels with itself twice
- (x + 2) appears once in a numerator and once in a denominator — cancel
After canceling:
(1)/(x + 1) × (x - 1)/(x + 4)
Step 3: Multiply: (x - 1)/[(x + 1)(x + 4)]
Done. No further simplification possible.
What You Cannot Cancel
This trips up almost everyone. You can only cancel factors — whole terms that are multiplied together.
You cannot cancel terms connected by addition or subtraction.
❌ Wrong: (x + 2 + 5) / (x + 2) → you cannot cancel the (x + 2) pieces
✓ Correct: First combine like terms: (x + 7) / (x + 2) → now you see there's nothing to cancel
Also remember: you cannot cancel across fractions. Each rational expression is its own unit. Only cross-cancel factors that appear in different numerators and denominators within the product.
Domain Restrictions
Every variable value that makes a denominator equal to zero must be excluded from your answer. These are your restrictions.
Before you cancel anything, identify values that make any denominator zero. These are values the original expression was undefined for, and they remain excluded even after simplification.
Example: If your original problem has (x + 1) in a denominator, then x ≠ -1. This restriction stays even if you cancel that factor later.
Quick Comparison: Factoring vs. Multiplying First
| Method | Process | Result |
|---|---|---|
| Multiply first, then simplify | Multiply all numerators, multiply all denominators, then try to factor the result | Often produces large polynomials that are harder to factor |
| Factor first, then cancel | Factor everything, cancel common factors, then multiply | Produces simpler, factored form with less work |
Always use the second method. It's faster and less error-prone.
Getting Started: Your Checklist
Before multiplying any rational expressions:
- ✓ Identify all denominators in the problem
- ✓ Note any values that make denominators zero (these are your restrictions)
- ✓ Factor every numerator and denominator completely
- ✓ Cancel any factor that appears in both a numerator and denominator
- ✓ Multiply the remaining numerators together
- ✓ Multiply the remaining denominators together
- ✓ State your final answer with the original restrictions
Common Mistakes to Avoid
- Trying to cancel before factoring — it doesn't work. Factor first, always.
- Canceling terms instead of factors — you can't cancel across addition or subtraction.
- Forgetting to state restrictions — the simplified answer might look fine, but the original expression was undefined for certain values.
- Not fully factoring — if you miss a factor, you miss a cancellation. Double-check your factoring.
- Multiplying everything out prematurely — you'll just have to factor it again to simplify.
Practice Problem
Multiply and simplify: (2x² - 8x)/(x² - 4) × (x² + 5x + 6)/(x² + 2x)
Answer below when you're ready to check.
Answer: 2(x + 3)/(x(x + 2)), with restrictions x ≠ 0, x ≠ -2, x ≠ 2
Did you get it? If not, go back and factor each polynomial again. Find the common factors. Cancel them. Multiply what's left.