Factoring Polynomials

Introduction to Factoring Polynomials

Factoring is the reverse of multiplying. When you multiply (x + 3)(x − 2), you get x² + x − 6. Factoring means starting with x² + x − 6 and figuring out that it came from (x + 3)(x − 2). This skill is essential for solving polynomial equations and simplifying rational expressions.

Different polynomial forms require different factoring techniques. This guide covers the four main methods you'll encounter: greatest common factor (GCF), difference of squares, trinomial factoring, and factoring by grouping. Recognizing which pattern you're dealing with is half the battle.

Method I: Greatest Common Factor (GCF)

The GCF method should always be your first check. Every term in the polynomial might share a common factor — a number, a variable, or both. Factor it out first, then check if what remains can be factored further.

Step-by-Step Process:

1. Find the greatest common factor of all terms (numbers and variables)
2. Write the GCF outside parentheses
3. Divide each term by the GCF and write the results inside parentheses

Worked Example: Factor 6x² + 9x.

• GCF of 6 and 9 is 3. GCF of x² and x is x.
• Combined GCF: 3x.
• 6x² ÷ 3x = 2x, and 9x ÷ 3x = 3.
• Result: 3x(2x + 3).

Another Example: Factor 12x³ − 8x² + 4x.

GCF is 4x. Result: 4x(3x² − 2x + 1).

Method II: Difference of Squares

A difference of squares takes the form a² − b². This pattern factors neatly as (a + b)(a − b). Recognizing perfect squares is the key.

Pattern: a² − b² = (a + b)(a − b)

Worked Example: Factor x² − 16.

• x² is a perfect square: (x)²
• 16 is a perfect square: (4)²
• Result: (x + 4)(x − 4)

Another Example: Factor 4x² − 25.

• 4x² = (2x)², 25 = (5)²
• Result: (2x + 5)(2x − 5)

Note: A sum of squares (a² + b²) does not factor over real numbers. x² + 9 cannot be factored further.

Method III: Factoring Trinomials

Trinomials of the form x² + bx + c can often be factored as (x + m)(x + n), where m and n are numbers that multiply to give c and add to give b. Finding these numbers is the core challenge.

Step-by-Step Process:

1. Identify b (coefficient of x) and c (constant term)
2. Find two numbers that multiply to c and add to b
3. Write the factored form: (x + m)(x + n)

Worked Example: Factor x² + 7x + 12.

• Need two numbers that multiply to 12 and add to 7.
• Pairs of factors of 12: (1, 12), (2, 6), (3, 4)
• 3 + 4 = 7, so m = 3 and n = 4.
• Result: (x + 3)(x + 4)

Example with Negative: Factor x² − 5x + 6.

• Need two numbers that multiply to 6 and add to −5.
• Both must be negative: (−2)(−3) = 6 and −2 + (−3) = −5.
• Result: (x − 2)(x − 3)

Example with Mixed Signs: Factor x² + 2x − 15.

• Need numbers that multiply to −15 and add to 2.
• One positive, one negative: 5 and −3 work (5 × −3 = −15, 5 + −3 = 2).
• Result: (x + 5)(x − 3)

Method IV: Factoring by Grouping

When a polynomial has four or more terms, grouping can help. Split the terms into pairs, factor each pair separately, and then look for a common factor in the results.

Step-by-Step Process:

1. Group terms into pairs that share a common factor
2. Factor out the GCF from each pair
3. Look for a common binomial factor
4. Factor out the common binomial

Worked Example: Factor x³ + 2x² − 3x − 6.

1. Group: (x³ + 2x²) + (−3x − 6)
2. Factor each pair: x²(x + 2) − 3(x + 2)
3. Common factor: (x + 2)
4. Result: (x² − 3)(x + 2)

Essential Factoring Reference Table

This table summarizes when to use each method:

Method	When to Use	Pattern	Key Insight
GCF	All terms share a factor	ax + bx = x(a + b)	Always check first
Difference of Squares	Two terms, both perfect squares	a² − b² = (a+b)(a−b)	Must be subtraction
Trinomials	Three terms, leading coeff = 1	x² + bx + c	Find factors of c that add to b
Grouping	Four or more terms	Pair and factor	Look for common binomial

Combining Methods

Many polynomials require multiple factoring steps. A common pattern: factor out the GCF first, then apply another method to what remains.

Worked Example: Factor 3x² − 27.

1. First, factor out GCF of 3: 3(x² − 9)
2. Then, recognize x² − 9 as difference of squares
3. Result: 3(x + 3)(x − 3)

Common Mistakes and How to Avoid Them

• Skipping the GCF check: Always look for a common factor first. Factoring x² − 9 is easy, but factoring 4x² − 36 requires pulling out 4 first.
• Sign errors with trinomials: When the constant term is negative, one factor will be positive and one negative. Double-check which number goes where.
• Incomplete factoring: After factoring, check that each factor is fully simplified. 4x² − 36 factors to 4(x² − 9), but x² − 9 factors further to 4(x + 3)(x − 3).
• Confusing sum and difference of squares: x² − 9 factors; x² + 9 does not. Remember that difference of squares requires subtraction.

Summary and Key Takeaways

Factoring polynomials comes down to pattern recognition. Check for a GCF first, then identify whether you're dealing with a difference of squares, a trinomial, or an expression that needs grouping. Practice each method until you can spot the pattern quickly, and always verify your answer by multiplying the factors back together.

With these four techniques, you can factor nearly any polynomial you'll encounter in Algebra 1 and 2. The key is developing an instinct for which method to try first.

For automated calculations and verification, use our Factoring Calculator to check your work and gain additional practice.