How to Factor Polynomials?
Knowing how to factor polynomials is a basis of Algebra 1 and beyond. Factoring undoes multiplication and lets you solve equations, simplify expressions, and analyze graphs. This page covers the most important factoring strategies, greatest common factor (GCF), difference of squares, perfect square trinomials, and factoring by grouping, with clear examples and two comprehensive video lessons.
Factor Polynomials: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Organize by degreeWrite terms from highest power to lowest power.
- Look for structureTry GCF, special products, grouping, or division depending on the expression.
- Check with featuresZeros, multiplicity, and end behavior should agree with your algebra.
Worked examples
Combine like terms
- Group x squared terms.
- Group x terms.
- Combine each group.
Factor a difference of squares
- Recognize a squared term minus a squared term.
- Use a^2 – b^2.
- Write conjugate factors.
Try one before moving on
Factor Polynomials: pop-up practice
What Does It Mean to Factor a Polynomial?
Factoring a polynomial means rewriting it as a product of simpler expressions (its factors) whose product equals the original polynomial. For example, \(\color{blue}{6x^{3} + 9x^{2} = 3x^{2}(2x + 3)}\). You can verify any factoring answer by multiplying the factors back together.
Strategy 1: Factor Out the Greatest Common Factor (GCF)
Always look for a GCF first. The GCF is the largest monomial that divides evenly into every term.
- \(\color{blue}{6x^{3} + 9x^{2}}\): GCF = \(\color{blue}{3x^{2}}\) → \(\color{blue}{3x^{2}(2x + 3)}\)
- \(\color{blue}{4x^{2} – 12x}\): GCF = \(\color{blue}{4x}\) → \(\color{blue}{4x(x – 3)}\)
Strategy 2: Difference of Two Squares
If a binomial has the form \(\color{blue}{a^{2} – b^{2}}\), it factors as \(\color{blue}{(a + b)(a – b)}\).
- \(\color{blue}{x^{2} – 16 = (x + 4)(x – 4)}\)
- \(\color{blue}{9x^{2} – 25 = (3x + 5)(3x – 5)}\)
Strategy 3: Perfect Square Trinomial
Recognize \(\color{blue}{a^{2} + 2\text{ ab } + b^{2} = (a + b)^{2}}\) and \(\color{blue}{a^{2} – 2\text{ ab } + b^{2} = (a – b)^{2}}\).
- \(\color{blue}{x^{2} + 6x + 9 = (x + 3)^{2}}\)
Strategy 4: Factoring Trinomials (ax² + \(\color{blue}{\text{ bx } + c}\))
For \(\color{blue}{a = 1}\), find two numbers with product c and sum b. For a > 1, use the AC method (see the Factoring Trinomials page).
Strategy 5: Factoring by Grouping
For four-term polynomials, group the terms in pairs, factor each pair, then factor out the common binomial.
- \(\color{blue}{x^{3} + 2x^{2} + 3x + 6}\) → \(\color{blue}{x^{2}(x + 2) + 3(x + 2)}\) → \(\color{blue}{(x^{2} + 3)(x + 2)}\)
Step-by-Step Summary
- Factor out the GCF first (always).
- Count the remaining terms: 2 terms → difference of squares or sum/difference of cubes; 3 terms → trinomial; 4 terms → grouping.
- Apply the appropriate strategy.
- Check each factor to see if it can be factored further.
- Verify by multiplying back.
Watch: How to Factor Polynomials (Video Lesson)
The Organic Chemistry Tutor covers the most common factoring strategies clearly and efficiently:
Factoring Polynomials — Worked Examples
Example 1: Factor \(\color{blue}{6x^{3} + 9x^{2}}\).
GCF = \(\color{blue}{3x^{2}}\). Divide each term: \(\color{blue}{6x^{3} \div 3x^{2} = 2x}\); \(\color{blue}{9x^{2} \div 3x^{2} = 3}\).
Answer: \(\color{blue}{3x^{2}(2x + 3)}\)
Example 2: Factor \(\color{blue}{x^{2} – 16}\).
Difference of squares: \(\color{blue}{a = x}\), \(\color{blue}{b = 4}\).
Answer: \(\color{blue}{(x + 4)(x – 4)}\)
Example 3: Factor \(\color{blue}{x^{2} + 6x + 9}\).
Perfect square: \(\color{blue}{\sqrt{9} = 3}\); middle term \(\color{blue}{2(x)(3) = 6x}\) matches.
Answer: \(\color{blue}{(x + 3)^{2}}\)
Example 4: Factor \(\color{blue}{x^{3} – x^{2} – 6x}\).
\(\color{blue}{\text{ GCF } = x}\): \(\color{blue}{x(x^{2} – x – 6)}\). Factor the trinomial: product = −6, sum = −1 → −3 and 2.
Answer: \(\color{blue}{x(x – 3)(x + 2)}\)
More Factoring Strategies: Comprehensive Video
This in-depth video covers GCF, AC method, grouping, and more, great for a full review:
Exercises for Factoring Polynomials
Factor each polynomial completely.
- \(\color{blue}{8x^{2} + 12x}\)
- \(\color{blue}{x^{2} – 25}\)
- \(\color{blue}{x^{2} + 6x + 9}\)
- \(\color{blue}{2x^{3} – 8x}\)
- \(\color{blue}{x^{3} – x^{2} – 6x}\)
- \(\color{blue}{3x^{2} – 12}\)
Answers
- \(\color{blue}{4x(2x + 3)}\)
- \(\color{blue}{(x + 5)(x – 5)}\)
- \(\color{blue}{(x + 3)^{2}}\)
- \(\color{blue}{2x(x + 2)(x – 2)}\)
- \(\color{blue}{x(x – 3)(x + 2)}\)
- \(\color{blue}{3(x + 2)(x – 2)}\)
Free Factoring Polynomials Worksheet
Ready to practice on your own? Download our free Factoring Polynomials worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again, steady practice is the surest way to master Factoring Polynomials before a quiz or test.
Download Greatest Common Factor and GCF Factoring Worksheet
Frequently Asked Questions
What is the easiest way to factor polynomials?
Always start by pulling out the greatest common factor, then look at how many terms are left. Two terms suggest a difference of squares, three terms suggest a trinomial or perfect square, and four terms suggest grouping. For example, 3x² + 12x = 3x(x + 4) after taking out the GCF 3x. Checking by multiplying the factors back together catches most mistakes.
How do you factor polynomials with 4 terms?
Use factoring by grouping. Split the polynomial into two pairs, factor each pair, and look for a shared binomial. For x³ + 3x² + 2x + 6, the first pair gives x²(x + 3) and the second gives 2(x + 3). The common factor is (x + 3), so the result is (x + 3)(x² + 2). If the binomials do not match, try rearranging the terms.
How do you factor polynomials when a is not 1?
Use the ac method. For 2x² + 7x + 3, multiply a × c = 2 × 3 = 6, then find two numbers that multiply to 6 and add to 7: 6 and 1. Rewrite the middle term: 2x² + 6x + x + 3. Group to get 2x(x + 3) + 1(x + 3), which factors as (2x + 1)(x + 3).
How do you factor polynomials with 2 terms?
Check for a GCF first, then for a difference of squares, a² − b² = (a − b)(a + b). For 9x² − 16, both terms are squares, (3x)² and 4², so it factors as (3x − 4)(3x + 4). A sum of squares such as x² + 4 does not factor over the real numbers, so it stays as it is.
Is factoring polynomials Algebra 1 or Algebra 2?
Both. Students usually learn to factor out a GCF, factor trinomials, use special products and factor by grouping in Algebra 1, often in 8th or 9th grade. Algebra 2 builds on those skills with higher-degree polynomials, the factor theorem and polynomial division. A strong Algebra 1 foundation in factoring makes the Algebra 2 work much smoother.
How do you factor polynomials completely?
Keep factoring until no factor can be broken down further. Begin with the GCF, then check every remaining factor for another pattern. For 2x³ − 18x, the GCF is 2x, leaving 2x(x² − 9). Since x² − 9 is a difference of squares, the complete factorization is 2x(x − 3)(x + 3). Stopping after the GCF is a common reason answers are marked incomplete.
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