How to Factor Polynomials (With Examples)

Factoring polynomials means breaking down a polynomial into simpler expressions (called factors) that multiply together to give the original polynomial. This guide covers the five most common factoring methods with step-by-step examples.

Method 1: Greatest Common Factor (GCF)

Always start by checking if all terms share a common factor. Factor out the GCF first.

Example:Factor 6x² + 9x

Step 1: Find the GCF of 6x² and 9x

GCF of 6 and 9 = 3

GCF of x² and x = x

GCF = 3x

Step 2: Factor out 3x

6x² + 9x = 3x(2x + 3)

Answer: 3x(2x + 3)

Method 2: Difference of Squares

When you see a² - b² (two perfect squares with a minus sign), use this formula:

a² - b² = (a + b)(a - b)

Example:Factor x² - 16

Step 1: Identify: x² - 16 = x² - 4²

Step 2: Apply formula with a = x, b = 4

Answer: (x + 4)(x - 4)

Method 3: Perfect Square Trinomial

A trinomial that is a perfect square follows one of these patterns:

a² + 2ab + b² = (a + b)²

a² - 2ab + b² = (a - b)²

Example:Factor x² + 6x + 9

Step 1: Check: first term = x², last term = 9 = 3², middle term = 2(x)(3) = 6x &check;

Step 2: Apply pattern: a = x, b = 3

Answer: (x + 3)²

Method 4: Factoring by Grouping

For polynomials with four terms, group terms in pairs and factor out the common factor from each pair.

Example:Factor x³ + 2x² + 3x + 6

Step 1: Group into pairs

(x³ + 2x²) + (3x + 6)

Step 2: Factor out GCF from each group

x²(x + 2) + 3(x + 2)

Step 3: Factor out common binomial (x + 2)

Answer: (x + 2)(x² + 3)

Method 5: Trinomials ax² + bx + c

For quadratic trinomials, find two numbers that multiply to give ac and add to give b.

Example:Factor x² + 5x + 6

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

2 × 3 = 6 and 2 + 3 = 5 &check;

Step 2: Write as binomial factors

Answer: (x + 2)(x + 3)

Which Method to Use?

1. First, check for a GCF in all terms

2. Two terms? → Check difference of squares

3. Three terms? → Check perfect square trinomial or trial and error

4. Four terms? → Try grouping

5. After factoring, check each factor again to see if it can be factored further

Practice Factoring

Use our free scientific calculator to check your factoring by expanding the result and verifying it matches the original.

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