How to Factor Polynomials

What Does It Mean to Factor a Polynomial?

Factoring a polynomial means writing it as a product of simpler expressions. It's like "un-multiplying" – you're finding what expressions, when multiplied together, give you the original polynomial.

For example, factoring x² + 5x + 6 gives you (x + 2)(x + 3). You can check this by multiplying: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6.

The Four Main Factoring Methods

Different polynomials require different techniques. Here's how to decide which method to use:

Method	When to Use	Example
GCF	All terms share a common factor	6x² + 9x = 3x(2x + 3)
Difference of Squares	Two terms, both perfect squares, subtracted	x² - 9 = (x+3)(x-3)
Trinomial Factoring	Three terms, ax² + bx + c form	x² + 7x + 12 = (x+3)(x+4)
Grouping	Four terms, can pair and factor	x³ + 3x² + 2x + 6

Method 1: Greatest Common Factor (GCF)

Always check for a GCF first. Look for the biggest number and highest power of each variable that divides all terms.

Example: Factor 12x³ + 18x²

Step 1: Find GCF of coefficients 12 and 18 → GCF = 6

Step 2: Find GCF of x³ and x² → GCF = x²

Step 3: Combined GCF = 6x²

Step 4: Factor out 6x²:

12x³ + 18x² = 6x²(2x + 3)

Method 2: Difference of Squares

When you have two perfect squares with a minus sign between them, use this pattern:

a² - b² = (a + b)(a - b)

Example: Factor x² - 25

Step 1: Identify the squares: x² is (x)² and 25 is (5)²

Step 2: Apply the formula:

x² - 25 = (x + 5)(x - 5)

Example: Factor 4x² - 9

4x² = (2x)² and 9 = (3)²

4x² - 9 = (2x + 3)(2x - 3)

Method 3: Factoring Trinomials (ax² + bx + c)

This is the most common type. The approach depends on whether the leading coefficient (a) equals 1.

Case A: When a = 1 (x² + bx + c)

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

Example: Factor x² + 7x + 12

Step 1: Find numbers that add to 7 and multiply to 12

3 + 4 = 7 ✓ and 3 × 4 = 12 ✓

Step 2: Write the factors:

x² + 7x + 12 = (x + 3)(x + 4)

Example: Factor x² - 5x + 6

Need numbers that add to -5 and multiply to +6

-2 + (-3) = -5 ✓ and (-2) × (-3) = 6 ✓

x² - 5x + 6 = (x - 2)(x - 3)

Case B: When a ≠ 1 (ax² + bx + c)

Use the "AC method" – multiply a and c, find factors, then regroup.

Example: Factor 2x² + 7x + 3

Step 1: Find a × c = 2 × 3 = 6

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

1 + 6 = 7 ✓ and 1 × 6 = 6 ✓

Step 3: Split the middle term:

2x² + 6x + x + 3

Step 4: Factor by grouping:

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

Step 5: Factor out (x + 3):

2x² + 7x + 3 = (2x + 1)(x + 3)

Method 4: Factoring by Grouping

For four-term polynomials, pair terms and factor each pair.

Example: Factor x³ + 2x² + 3x + 6

Step 1: Group terms in pairs

(x³ + 2x²) + (3x + 6)

Step 2: Factor each pair

x²(x + 2) + 3(x + 2)

Step 3: Factor out the common factor (x + 2)

x³ + 2x² + 3x + 6 = (x + 2)(x² + 3)

Step-by-Step Strategy

When you see a polynomial, follow this order:

1. Check for GCF first – factor out any common factor
2. Count the terms:
   • 2 terms → Try difference of squares
   • 3 terms → Try trinomial factoring
   • 4 terms → Try grouping
3. Check if factors can be factored further
4. Verify by multiplying back

Common Mistakes to Avoid

• Skipping the GCF: Always check for a GCF first, even if it's just a number.
• Wrong signs: Watch the signs carefully. x² - 9 = (x+3)(x-3), not (x-3)(x-3).
• Stopping too soon: After factoring, check if each factor can be factored further.
• Forgetting to check: Multiply your answer to verify it matches the original.

Verifying Your Answer

Always multiply your factored form to check:

Check: (x + 3)(x - 2) = x² - 2x + 3x - 6 = x² + x - 6

If you started with x² + x - 6, your factoring is correct!

Summary

Factoring polynomials comes down to recognizing patterns and choosing the right method:

• GCF: Always check first
• Difference of squares: a² - b² = (a+b)(a-b)
• Trinomials: Find numbers that add and multiply correctly
• Grouping: Pair, factor, then factor again

Practice each method until you can recognize which one to use at a glance.

For quick calculations, try our Factoring Calculator to check your work and explore more polynomials.