How to Factor Polynomials Step by Step

Factoring polynomials is one of the most important skills in algebra. Whether you are simplifying expressions, solving equations, or working through calculus problems later on, factoring shows up everywhere. This guide walks through every major factoring technique with worked examples so you can handle any polynomial that comes your way.

If you are looking for a broader overview of algebra topics, check out our Algebra Guide.

Why Factoring Matters

Factoring rewrites a polynomial as a product of simpler expressions. This lets you find the zeros of a function, simplify rational expressions, and solve equations that would otherwise be difficult to work with. Think of it as the reverse of distributing — instead of expanding, you are compressing.

Method 1: Greatest Common Factor (GCF)

Always start here. Before trying any other technique, pull out the largest factor shared by every term.

Example 1: Factor 12x^3 + 18x^2 - 6x

1. Find the GCF of the coefficients: GCF of 12, 18, and 6 is 6.
2. Find the lowest power of x present in every term: x^1.
3. Factor it out: 6x(2x^2 + 3x - 1)

If you pulled out the GCF and the expression inside the parentheses still has more than one term, check whether it can be factored further using the methods below.

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Method 2: Factoring Trinomials (ax^2 + bx + c)

This is the technique students use most often. There are two cases depending on whether the leading coefficient is 1 or something else.

Case A: Leading coefficient is 1 (x^2 + bx + c)

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

Example 2: Factor x^2 + 7x + 12

• You need two numbers that multiply to 12 and add to 7.
• 3 and 4 work: 3 * 4 = 12, 3 + 4 = 7.
• Result: (x + 3)(x + 4)

Case B: Leading coefficient is not 1 (ax^2 + bx + c)

Use the AC method: multiply a * c, find two numbers that multiply to that product and add to b, then split the middle term and factor by grouping.

Example 3: Factor 6x^2 + 11x + 3

1. a c = 6 3 = 18.
2. Find two numbers that multiply to 18 and add to 11: 9 and 2.
3. Rewrite the middle term: 6x^2 + 9x + 2x + 3.
4. Group: (6x^2 + 9x) + (2x + 3).
5. Factor each group: 3x(2x + 3) + 1(2x + 3).
6. Factor out the common binomial: (3x + 1)(2x + 3)

Method 3: Difference of Squares

This pattern applies when you have two perfect squares separated by a minus sign.

Formula: a^2 - b^2 = (a + b)(a - b)

Example 4: Factor 25x^2 - 49

• 25x^2 = (5x)^2 and 49 = 7^2.
• Result: (5x + 7)(5x - 7)

Note: a sum of squares (a^2 + b^2) does not factor over the real numbers.

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Method 4: Factoring by Grouping

When you have four terms, grouping is usually the way to go. Pair the terms, factor each pair, and look for a common binomial.

Example 5: Factor x^3 + 2x^2 + 5x + 10

1. Group: (x^3 + 2x^2) + (5x + 10).
2. Factor each group: x^2(x + 2) + 5(x + 2).
3. Factor out (x + 2): (x + 2)(x^2 + 5)

If the first grouping does not produce a common binomial, try rearranging the terms.

Method 5: Sum and Difference of Cubes

These formulas are worth memorizing:

• Sum of cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2)
• Difference of cubes: a^3 - b^3 = (a - b)(a^2 + ab + b^2)

A handy mnemonic: SOAP — Same sign, Opposite sign, Always Positive. The first factor shares the sign from the original expression, the second term in the trinomial takes the opposite sign, and the last term is always positive.

Example 6: Factor 8x^3 - 27

• 8x^3 = (2x)^3 and 27 = 3^3.
• Using the difference of cubes formula: (2x - 3)(4x^2 + 6x + 9)

Example 7: Factor x^3 + 125

• x^3 = (x)^3 and 125 = 5^3.
• Using the sum of cubes formula: (x + 5)(x^2 - 5x + 25)

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Putting It All Together: A Strategy

When you see a polynomial to factor, follow this order:

1. GCF first. Always.
2. Count the terms. Two terms — check for difference of squares or sum/difference of cubes. Three terms — try trinomial factoring. Four terms — try grouping.
3. Check your answer by multiplying the factors back together. The result should match the original expression.
4. Look inside the factors. Sometimes a factor can be factored again (for example, a difference of squares hiding inside a trinomial result).

Common Mistakes to Avoid

• Forgetting to factor out the GCF before moving on to other methods.
• Mixing up the sum of cubes and difference of cubes formulas (remember SOAP).
• Stopping too early — always check whether the factors themselves can be factored further.
• Sign errors when splitting the middle term in the AC method.

FAQ

What if I cannot find two numbers that work for a trinomial?

Not every trinomial factors neatly over the integers. If no pair of integers multiplies to ac and adds to b, the trinomial is either prime (cannot be factored with integer coefficients) or requires the quadratic formula to find its roots.

Is there a quick way to check my factoring?

Yes. Multiply your factors back together and confirm you get the original expression. You can also use the free math checker at Math.Photos to verify your work.

When do I use grouping versus the AC method?

They are closely related. The AC method for trinomials actually uses grouping as its final step. Grouping on its own is typically used when you start with four terms rather than three.

Do these methods work for polynomials of degree higher than 2?

Absolutely. GCF, grouping, and the cube formulas all apply to higher-degree polynomials. For degree 3 and above, you may also need synthetic division or the rational root theorem, but factoring techniques covered here remain your first line of attack.

For more algebra walkthroughs, head back to our Algebra Guide.