Curriculum fit: US Algebra 1. Aligned to Common Core HSA-SSE.A.2 (use the structure of an expression to rewrite it) and HSA-APR.B.3 (zeros of polynomials). Check your own state standards and course.
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What Is It?
Factoring means writing an expression as a product. It is multiplying out in reverse: x² + 5x + 6 becomes (x + 2)(x + 3).
Always follow the same order. First take out any greatest common factor (GCF). Then look at the number of terms and choose the method: two terms, think difference of squares; three terms, find two numbers; four terms, try grouping.
You can always check by multiplying your answer out. If you get the original expression back, it is right.
Key Formulas and Ideas
- Greatest common factor: take out the largest factor shared by every term.
- x² + bx + c = (x + p)(x + q) where p × q = c and p + q = b.
- ax² + bx + c: find two numbers that multiply to a × c and add to b, split the middle term, then factor by grouping.
- Difference of squares: a² − b² = (a − b)(a + b).
- Perfect square trinomial: a² ± 2ab + b² = (a ± b)².
Worked Examples
Simple Trinomial
Factor x² + 7x + 12.
- Find two numbers that multiply to 12 and add to 7.
- 3 × 4 = 12 and 3 + 4 = 7.
- Write (x + 3)(x + 4). Check: x² + 4x + 3x + 12 = x² + 7x + 12.
✓ Answer (x + 3)(x + 4)
Trinomial With a Leading Coefficient
Factor 6x² + 11x + 3.
- Multiply a × c = 6 × 3 = 18. Find two numbers that multiply to 18 and add to 11: 9 and 2.
- Split the middle term: 6x² + 9x + 2x + 3.
- Group: 3x(2x + 3) + 1(2x + 3) = (3x + 1)(2x + 3).
✓ Answer (3x + 1)(2x + 3)
GCF Then Difference of Squares
Factor 3x³ − 12x completely.
- Take out the GCF 3x: 3x(x² − 4).
- x² − 4 is a difference of squares: (x − 2)(x + 2).
- Write all the factors together.
✓ Answer 3x(x − 2)(x + 2)
Common Mistakes to Avoid
- Forgetting to take out the GCF first, then stopping before the expression is fully factored.
- Sign mistakes: x² − 5x + 6 needs two negative numbers, (x − 2)(x − 3).
- Trying to factor a sum of squares such as x² + 9 with real numbers. It does not factor.
- Not checking by multiplying out.
Practice Questions
Frequently Asked Questions
How do I know when a polynomial is fully factored?
It is fully factored when none of the remaining factors can be factored further and no common factor is left. Always check for a GCF first and last.
Why factor at all?
Factoring helps you solve quadratic equations. If (x − 2)(x + 5) = 0, then x − 2 = 0 or x + 5 = 0, so x = 2 or x = −5.
What if a trinomial will not factor?
Some quadratics cannot be factored with whole numbers. Use the quadratic formula or completing the square instead.
Keep Going With These Guides
Solving Quadratic Equations: Factorising and the Formula
Solve quadratics by factorising and with the quadratic formula, using worked examples. Covers GCSE quadratic equations and Algebra 1 and 2 methods.
GCSEA-LevelHigh SchoolQuadratic Graphs: Roots, Turning Points and Symmetry
Sketch parabolas and find the roots, y-intercept, line of symmetry and turning point (vertex) of a quadratic, with step-by-step worked examples.
GCSEA-LevelHigh SchoolSimplifying Algebraic Expressions: Collecting Like Terms
Collect like terms, multiply and divide algebraic terms and simplify expressions with brackets, with step-by-step examples and practice questions.
KS3GCSEMiddle SchoolHigh SchoolAlgebra 1 Exponent Rules With Worked Examples
The laws of exponents in Algebra 1: product, quotient and power rules, zero and negative exponents, with step-by-step examples and common mistakes.
Middle SchoolHigh SchoolSources and Review
- Common Core: High School Algebra, Seeing Structure in Expressions
- Common Core: High School Algebra, Arithmetic with Polynomials
Reviewed by StudyMorrow editorial review on 6 Nov 2026.