How to Factor Polynomials (With Examples)

Factoring polynomials is one of the most important skills in algebra. It helps simplify expressions, solve equations, and understand mathematical relationships more clearly. Factoring is widely used in solving quadratic equations, graphing functions, and even in higher-level mathematics like calculus.

In simple terms, factoring a polynomial means breaking it into smaller expressions that multiply together to form the original polynomial.

For example:

x² − 9 can be factored as
(x − 3)(x + 3)

What Does It Mean to Factor a Polynomial?

Factoring means rewriting a polynomial as a product of simpler polynomials or expressions.

Example:

6x + 9

Both terms have a common factor of 3.

So it can be written as:

3(2x + 3)

Here, 3(2x + 3) is the factored form of 6x + 9.

Methods to Factor Polynomials

There are several methods used to factor polynomials depending on the type of expression.

1. Factoring Out the Greatest Common Factor (GCF)

This is the first method you should always check.

Steps:

1. Find the greatest common number or variable in all terms.
2. Divide each term by that common factor.
3. Write the common factor outside the bracket.

Example 1

Factor: 8x² + 12x

Step 1: GCF of 8x² and 12x is 4x

Step 2: Divide each term by 4x

8x² ÷ 4x = 2x
12x ÷ 4x = 3

Step 3: Write the factored form

4x(2x + 3)

2. Factoring Trinomials (Quadratic Polynomials)

This method is used for expressions like:

ax² + bx + c

Example 2

Factor: x² + 5x + 6

Step 1: Find two numbers that multiply to 6 and add to 5

Those numbers are 2 and 3

Step 2: Write the factors

(x + 2)(x + 3)

Example 3 (With Coefficient)

Factor: 2x² + 7x + 3

Step 1: Multiply a × c

2 × 3 = 6

Step 2: Find two numbers that multiply to 6 and add to 7

6 and 1

Step 3: Rewrite middle term

2x² + 6x + x + 3

Step 4: Group terms

(2x² + 6x) + (x + 3)

Step 5: Factor each group

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

Step 6: Final answer

(2x + 1)(x + 3)

3. Difference of Squares

This applies when you see:

a² − b²

Formula:

a² − b² = (a − b)(a + b)

Example 4

Factor: x² − 16

x² − 4²

= (x − 4)(x + 4)

4. Perfect Square Trinomials

These follow special patterns:

a² + 2ab + b² = (a + b)²
a² − 2ab + b² = (a − b)²

Example 5

Factor: x² + 6x + 9

Step 1: Recognize 9 = 3²
Step 2: Middle term 6x = 2 × x × 3

So the factor form is:

(x + 3)²

How to Know Which Method to Use

When factoring a polynomial, follow this order:

1. Always check for GCF first
2. Count the number of terms
   • Two terms → Check difference of squares
   • Three terms → Try trinomial factoring
3. Look for special patterns like perfect squares

Practical Examples of Factoring Polynomials

Example 6

Factor: 3x² − 12

Step 1: Take out GCF

3(x² − 4)

Step 2: Recognize difference of squares

3(x − 2)(x + 2)

Example 7

Factor: x² − x − 6

Step 1: Multiply a × c

1 × (−6) = −6

Step 2: Find numbers that multiply to −6 and add to −1

−3 and 2

Step 3: Final answer

(x − 3)(x + 2)

Tips to Factor Polynomials Easily

1. Always Look for GCF First

Many students forget this step and make the problem harder.

2. Practice Number Combinations

Be quick at finding two numbers that multiply and add correctly.

3. Memorize Special Formulas

Remember:

• a² − b² = (a − b)(a + b)
• a² + 2ab + b² = (a + b)²
• a² − 2ab + b² = (a − b)²

4. Double Check by Multiplying

After factoring, multiply the brackets again to see if you get the original polynomial.

5. Organize Your Work

Write steps clearly, especially when grouping terms.