Methods of Factoring Polynomials- Techniques and Examples

What Factoring Polynomials Actually Is

Factoring polynomials means rewriting a polynomial as a product of simpler polynomials. That's it. You're breaking down something complex into pieces that multiply back together.

This isn't busywork. Factoring shows up everywhere—in solving equations, simplifying expressions, and finding roots. If you're working with quadratics or higher-degree polynomials, you need these techniques in your toolkit.

Greatest Common Factor (GCF)

This is the first thing you should always check. Find the largest factor that divides every term.

How to find GCF

Examples

Example 1: Factor 12x³ + 18x²

GCF of 12 and 18 is 6. GCF of x³ and x² is x². So the GCF is 6x².

12x³ + 18x² = 6x²(2x + 3)

Example 2: Factor 5a²b³ - 15ab² + 10a³b

GCF is 5ab².

5a²b³ - 15ab² + 10a³b = 5ab²(a²b - 3 + 2a²)

Always factor out the GCF first. It's not optional—it's the foundation everything else builds on.

Factoring by Grouping

Use this when you have four terms with no obvious GCF across all of them. The trick is grouping terms that share common factors.

Process

  1. Group the first two terms together, last two terms together
  2. Factor out the GCF from each group
  3. If the binomials match, factor them out

Example

Factor 3x³ + 6x² + 2x + 4

Group: (3x³ + 6x²) + (2x + 4)

Factor each: 3x²(x + 2) + 2(x + 2)

Factor out (x + 2): (x + 2)(3x² + 2)

Sometimes you need to rearrange terms first. If grouping doesn't work immediately, try a different grouping arrangement.

Factoring Trinomials

Trinomials have three terms, usually in the form ax² + bx + c. The approach depends on whether a = 1 or a ≠ 1.

When a = 1 (x² + bx + c)

Find two numbers that multiply to c and add to b.

Example: Factor x² + 7x + 12

Find two numbers that multiply to 12 and add to 7. Those numbers are 3 and 4.

x² + 7x + 12 = (x + 3)(x + 4)

Example: Factor x² - 5x + 6

Need numbers that multiply to 6 and add to -5. That's -2 and -3.

x² - 5x + 6 = (x - 2)(x - 3)

When a ≠ 1 (AC Method)

For ax² + bx + c where a ≠ 1, use the AC method:

  1. Multiply a and c (the "AC" product)
  2. Find two numbers that multiply to AC and add to b
  3. Split the middle term using those numbers
  4. Factor by grouping

Example: Factor 2x² + 7x + 3

AC = 2 × 3 = 6

Find numbers that multiply to 6 and add to 7: 6 and 1

2x² + 7x + 3 = 2x² + 6x + x + 3

Group: 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3)

= (x + 3)(2x + 1)

Box Method (Alternative)

Some students prefer the box method. Draw a 2×2 grid. Put ax² in the top-left, c in the bottom-right. Fill the remaining corners so the diagonals multiply correctly, then read off the factored form.

It works. It's visual. Use it if the AC method gives you trouble.

Difference of Squares

Recognize this pattern: a² - b² = (a + b)(a - b)

The key is identifying perfect squares within your expression.

Examples

Example 1: Factor x² - 16

x² is a perfect square (x)². 16 is 4².

x² - 16 = (x + 4)(x - 4)

Example 2: Factor 49a² - 25b²

49a² = (7a)². 25b² = (5b)².

49a² - 25b² = (7a + 5b)(7a - 5b)

Example 3: Factor x⁴ - 16

This is nested squares: (x²)² - 4². Apply the formula twice.

x⁴ - 16 = (x² + 4)(x² - 4)

= (x² + 4)(x + 2)(x - 2)

Remember: you cannot factor a sum of squares over the real numbers. a² + b² stays as is. Don't try to force it.

Perfect Square Trinomials

These are trinomials that come from squaring a binomial. Recognize them quickly:

Check the middle term. Is it exactly twice the product of the square roots of the first and last terms? If yes, you have a perfect square trinomial.

Example

Factor 9x² + 12x + 4

Check: √(9x²) = 3x, √4 = 2. Twice their product: 2(3x)(2) = 12x. Matches the middle term.

9x² + 12x + 4 = (3x + 2)²

Sum and Difference of Cubes

Less common but you need them for higher-degree polynomials.

Sum of cubes: a³ + b³ = (a + b)(a² - ab + b²)

Difference of cubes: a³ - b³ = (a - b)(a² + ab + b²)

Example

Factor 8x³ + 27

8x³ = (2x)³. 27 = 3³.

8x³ + 27 = (2x + 3)((2x)² - (2x)(3) + 3²)

= (2x + 3)(4x² - 6x + 9)

The quadratic factor (a² - ab + b² or a² + ab + b²) won't factor further over the real numbers. That's your final answer.

Method Comparison

Method Pattern/Form When to Use
GCF Common factor in all terms Always check this first
Grouping Four terms, no overall GCF Four-term expressions
Trinomials (a=1) x² + bx + c Two numbers to c, sum to b
AC Method ax² + bx + c, a ≠ 1 Harder trinomials
Difference of Squares a² - b² Two perfect squares subtracted
Perfect Square a² ± 2ab + b² Middle term is 2√(first × last)
Cubes a³ ± b³ Two perfect cubes added or subtracted

Getting Started: A Practical Approach

Don't try to guess which method to use. Work through this checklist in order:

  1. Check for GCF first. Factor it out completely before doing anything else.
  2. Count the terms. Two terms? Check for squares or cubes. Three terms? Look for perfect square or use trinomial methods. Four terms? Try grouping.
  3. Look for recognizable patterns. Difference of squares, cubes, perfect squares—these have distinct forms.
  4. Count negative signs. They tell you whether binomial factors contain plus or minus.
  5. Multiply your answer back to verify it gives you the original polynomial.

Practice problem: Factor 2x³y + 8xy - 4x²y²

Step 1: GCF is 2xy.

2xy(x² + 4 - 2xy)

Step 2: Rearrange the trinomial inside: x² - 2xy + 4. Can this factor?

Check discriminant: (-2y)² - 4(1)(4) = 4y² - 16. This is negative unless y² > 4, so it doesn't factor nicely over integers. Your answer is 2xy(x² - 2xy + 4).

Common Mistakes to Avoid