How to Factor an Expression- Algebra Tutorial
What Factoring Actually Is
Factoring is breaking down a complex algebraic expression into simpler parts that multiply together to give you the original expression. That's it. No fancy definitions. You take something like x² + 5x + 6 and rewrite it as (x + 2)(x + 3).
Why bother? Because factored form makes solving equations way easier. Instead of using the quadratic formula on x² + 5x + 6 = 0, you can see immediately that x = -2 or x = -3.
The Methods You'll Actually Use
Most problems you'll encounter fall into a handful of patterns. Learn these, and you can handle 90% of factoring problems.
1. Factoring Out the Greatest Common Factor (GCF)
This is always the first step. Look at every term and pull out what they share.
Example:
12x³ + 18x² = 6x²(2x + 3)
Check your work by distributing: 6x² × 2x = 12x³ and 6x² × 3 = 18x². If it matches, you're correct.
2. Factoring Trinomials
For expressions like ax² + bx + c, you need two numbers that multiply to ac and add to b.
Example:
x² + 7x + 12
Find two numbers that multiply to 12 and add to 7. That's 3 and 4.
Answer: (x + 3)(x + 4)
When the leading coefficient isn't 1, you need to factor by grouping. Break the middle term using your two numbers, then group.
Example:
2x² + 5x - 3
Multiply 2 × (-3) = -6. Find numbers that multiply to -6 and add to 5. That's 6 and -1.
Rewrite: 2x² + 6x - x - 3
Group: (2x² + 6x) + (-x - 3)
Factor each: 2x(x + 3) - 1(x + 3)
Final answer: (2x - 1)(x + 3)
3. Difference of Squares
Pattern: a² - b² = (a + b)(a - b)
Examples:
- x² - 9 = (x + 3)(x - 3) ✓
- 4x² - 25 = (2x + 5)(2x - 5) ✓
- x⁴ - 16 = (x² + 4)(x² - 4) = (x² + 4)(x + 2)(x - 2) ✓
Only works for subtraction. Sum of squares doesn't factor over real numbers.
4. Perfect Square Trinomials
Pattern: a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)²
Example:
x² + 6x + 9
Square root of x² is x. Square root of 9 is 3. Twice their product is 2 × x × 3 = 6x, which matches the middle term.
Answer: (x + 3)²
5. Sum and Difference of Cubes
Less common but you'll see these:
- a³ + b³ = (a + b)(a² - ab + b²)
- a³ - b³ = (a - b)(a² + ab + b²)
Example:
x³ - 27
This is x³ - 3³. Apply the difference of cubes formula.
Answer: (x - 3)(x² + 3x + 9)
Quick Reference: Factoring Methods
| Method | Pattern | Example |
|---|---|---|
| GCF | Pull out common factor | 6x²(2x + 3) |
| Trinomial (a=1) | Find numbers that multiply to c, add to b | x² + 5x + 6 = (x+2)(x+3) |
| Trinomial (a≠1) | Factor by grouping | 2x² + 5x - 3 = (2x-1)(x+3) |
| Difference of Squares | a² - b² | x² - 9 = (x+3)(x-3) |
| Perfect Square Trinomial | a² ± 2ab + b² | x² + 6x + 9 = (x+3)² |
| Sum of Cubes | a³ + b³ | x³ + 8 = (x+2)(x²-2x+4) |
| Difference of Cubes | a³ - b³ | x³ - 8 = (x-2)(x²+2x+4) |
How to Factor: Step-by-Step
When you see an expression to factor, follow this order:
Step 1: Check for a GCF
Always. Pull out the biggest number and variable that divides every term. If there isn't one, move on.
Step 2: Count the Terms
- 2 terms → Difference of squares, sum/difference of cubes, or GCF
- 3 terms → Trinomial or perfect square
- 4 terms → Try grouping
Step 3: Apply the Pattern
Match what you have to the patterns above. If it doesn't fit, try another method.
Step 4: Check Your Answer
Multiply your factors back out. Does it give you the original expression? If yes, you're done. If no, go back and find your mistake.
Common Mistakes That Cost You Points
- Forgetting to check for a GCF first — This is the most common error. Always look for a common factor before trying other methods.
- Misidentifying the pattern — A sum of squares doesn't factor. Stop trying to force it.
- Sign errors in the binomial — When factoring trinomials, watch those signs carefully.
- Not distributing to check your answer — This takes 5 seconds and catches most mistakes.
- Over-factoring — Sometimes an expression is already fully factored. Don't keep going.
Practice Problems
Try these before checking answers:
- x² - 4
- 3x² + 12x
- x² + 8x + 15
- 2x² + 7x + 3
- x³ + 64
Answers:
- (x + 2)(x - 2)
- 3x(x + 4)
- (x + 3)(x + 5)
- (2x + 1)(x + 3)
- (x + 4)(x² - 4x + 16)
When You're Stuck
If nothing seems to fit, the expression might not factor over real numbers. That's fine. Some expressions require the quadratic formula or are prime. You can't force a factorization that doesn't exist.
Also, make sure you copied the problem correctly. Transcribing errors happen constantly and will waste your time.
Factoring is a skill. The more you practice, the faster you recognize patterns. Start with simple expressions, work your way up. You'll get there.