How to Factor an Equation- Step-by-Step Guide
What Is Factoring?
Factoring is breaking down a complex expression into simpler pieces that multiply together to give you the original equation. It's the reverse of expanding parentheses.
Instead of 3x + 6, you factor it into 3(x + 2). Same math, different form. This skill shows up constantly in algebra, calculus, and solving real-world problems.
Factoring Out the Greatest Common Factor (GCF)
This is the easiest type of factoring. Find the largest number or variable that divides evenly into every term.
How to Find the GCF
- Look at the coefficients — find their greatest common divisor
- Look at the variables — use the smallest exponent that appears in every term
- Multiply those together
Example: 12x³ + 18x²
- GCF of 12 and 18 is 6
- Smallest x power is x²
- GCF = 6x²
Factor it: 6x²(2x + 3)
That's it. Pull out what they share, write what's left over in the parentheses.
Factoring Trinomials: x² + bx + c
These look like ax² + bx + c where "a" equals 1. The goal is two binomials that multiply to give you the original.
The Simple Method
For x² + 5x + 6:
- You need two numbers that multiply to give 6 (the constant) and add to give 5 (the coefficient of x)
- 2 and 3 work: 2 × 3 = 6, 2 + 3 = 5
- Write it as (x + 2)(x + 3)
Check by FOILing: x·x + x·3 + 2·x + 2·3 = x² + 5x + 6 ✓
When "c" Is Negative
For x² - x - 12:
You need numbers that multiply to -12 and add to -1.
-4 and 3 work: -4 × 3 = -12, -4 + 3 = -1.
Answer: (x - 4)(x + 3)
One trick: if c is negative, the signs in your binomials will be different. If c is positive, the signs match the sign of b.
The Difference of Squares
Some expressions are specifically two perfect squares subtracted from each other:
a² - b² = (a + b)(a - b)
That's the pattern. Spot it, apply it.
Examples:
- x² - 9 = x² - 3² = (x + 3)(x - 3)
- 4x² - 25 = (2x)² - 5² = (2x + 5)(2x - 5)
- 16y⁴ - 1 = (4y²)² - 1² = (4y² + 1)(4y² - 1)
Sum of squares doesn't factor over real numbers. Don't waste time trying — it doesn't work.
Perfect Square Trinomials
These factor into a binomial squared:
- a² + 2ab + b² = (a + b)²
- a² - 2ab + b² = (a - b)²
Example: x² + 6x + 9
√x² = x, √9 = 3, and 2(x)(3) = 6x ✓
Answer: (x + 3)²
Factoring by Grouping
When there's no obvious GCF across all terms, try grouping. Split the expression into two parts, factor each part, then look for a common binomial.
Example: 2x³ + 3x² + 2x + 3
- Group: (2x³ + 3x²) + (2x + 3)
- Factor each: x²(2x + 3) + 1(2x + 3)
- Notice (2x + 3) appears in both
- Final answer: (2x + 3)(x² + 1)
Sum and Difference of Cubes
Less common but you need these formulas:
- a³ + b³ = (a + b)(a² - ab + b²)
- a³ - b³ = (a - b)(a² + ab + b²)
Example: x³ - 27
x³ - 27 = x³ - 3³ = (x - 3)(x² + 3x + 9)
Remember the signs: SOAP — Same, Opposite, Always Positive
Factoring Methods Comparison
| Method | Pattern | Example |
|---|---|---|
| GCF | Factor out shared term | 6x² + 9x = 3x(2x + 3) |
| Trinomials (a=1) | x² + bx + c | x² + 5x + 6 = (x+2)(x+3) |
| Difference of Squares | a² - b² | x² - 16 = (x+4)(x-4) |
| Perfect Square Trinomial | a² ± 2ab + b² | x² + 6x + 9 = (x+3)² |
| Sum/Difference of Cubes | a³ ± b³ | x³ - 8 = (x-2)(x²+2x+4) |
| Grouping | 4+ terms, no common GCF | 2x³ + 3x² + 2x + 3 |
How to Factor: Step-by-Step Process
When you see an expression to factor, work through this checklist:
- Check for a GCF first — always. Pull it out before doing anything else.
- Count the terms:
- 2 terms → difference of squares or cubes?
- 3 terms → trinomial pattern or perfect square?
- 4+ terms → try grouping
- Check the signs — positive c means same signs in binomials, negative c means opposite signs.
- Verify your answer — multiply back using FOIL or distribution. Does it match?
Practice Examples
1. Factor: 8x² - 32
GCF is 8: 8(x² - 4)
x² - 4 is a difference of squares: (x + 2)(x - 2)
Final answer: 8(x + 2)(x - 2)
2. Factor: 3x² + 11x + 6
Multiply a×c = 3×6 = 18. Find two numbers that multiply to 18 and add to 11.
9 and 2 work. Rewrite: 3x² + 9x + 2x + 6
Group: (3x² + 9x) + (2x + 6)
Factor: 3x(x + 3) + 2(x + 3)
Final answer: (3x + 2)(x + 3)
3. Factor: 5x³y + 10x²y² - 15xy³
GCF: 5xy is in every term
Factor out 5xy: 5xy(x² + 2xy - 3y²)
Now factor the trinomial inside: find numbers that multiply to -3 and add to 2.
3 and -1 work.
Final answer: 5xy(x + 3y)(x - y)
Common Mistakes to Avoid
- Forgetting to check for a GCF before trying other methods
- Getting the signs wrong when c is negative — the binomials need opposite signs
- Trying to factor a sum of squares (x² + 4) — it doesn't factor over real numbers
- Not verifying your answer by multiplying back
- Dropping a term when factoring out a GCF — make sure every term still appears
When You're Stuck
If none of the standard patterns work, the expression might be prime — meaning it can't be factored further using real numbers. That's a valid answer. Not every expression factors nicely.
Also check if you can factor using substitution. Sometimes x⁴ + 5x² + 6 looks hard until you notice (x²)² + 5(x²) + 6 and treat x² as a single variable.