Factoring Polynomials
9 min read · Updated September 26, 2026 · by the SolveCalcPro editorial team
Factoring is the reverse of multiplying: you rewrite a polynomial as a product of simpler ones. It is how you solve quadratic equations, simplify algebraic fractions and understand where graphs cross the axis. This guide walks through each method in the order you should try them.
Step 1: always look for a greatest common factor
Pull out any factor shared by every term. 6x² + 9x = 3x(2x + 3). And 4a³ − 8a² + 12a = 4a(a² − 2a + 3). Doing this first makes everything that follows easier.
Factoring trinomials with a leading 1
For x² + bx + c, find two numbers that multiply to c and add to b. For x² + 7x + 12, the numbers are 3 and 4, so (x + 3)(x + 4). For x² − 5x − 14, the numbers are −7 and 2 (product −14, sum −5), so (x − 7)(x + 2).
The ac method for other trinomials
For 2x² + 7x + 3, multiply a × c = 6 and find two numbers that multiply to 6 and add to 7: 1 and 6. Split the middle term: 2x² + x + 6x + 3, group as x(2x + 1) + 3(2x + 1), and factor out (2x + 1) to get (2x + 1)(x + 3).
Special patterns
| Pattern | Factors | Example |
|---|---|---|
| Difference of squares | a² − b² = (a − b)(a + b) | x² − 49 = (x − 7)(x + 7) |
| Perfect square trinomial | a² + 2ab + b² = (a + b)² | x² + 10x + 25 = (x + 5)² |
| Sum of cubes | a³ + b³ = (a + b)(a² − ab + b²) | x³ + 8 = (x + 2)(x² − 2x + 4) |
| Difference of cubes | a³ − b³ = (a − b)(a² + ab + b²) | x³ − 27 = (x − 3)(x² + 3x + 9) |
Note that a sum of squares, such as x² + 9, does not factor over the real numbers.
Factoring by grouping
For four terms, group in pairs: x³ + 2x² + 3x + 6 = x²(x + 2) + 3(x + 2) = (x² + 3)(x + 2).
Always check
Multiply your factors back out. (x − 7)(x + 2) = x² + 2x − 7x − 14 = x² − 5x − 14 ✓. Use the polynomial calculator to check, and the quadratic solver to confirm the roots.
Practice questions
Test yourself, then tap to check. Answers are calculated by the tools linked below.
- Multiply (x + 3)(x − 5).
Show answer
Product: x² − 2x − 15
- Solve x² − 7x + 10 = 0.
Show answer
x₁: 5 · x₂: 2
- Factor 252 into primes.
Show answer
Prime factorization: 2^2 × 3^2 × 7
Frequently asked questions
What is the first step in factoring?
Look for a greatest common factor and pull it out.
How do I know if a polynomial can be factored?
Try the methods in order. For a quadratic, the discriminant tells you: if b² − 4ac is a perfect square, it factors over the integers.
Can x² + 9 be factored?
Not over the real numbers. A sum of squares is prime.
How does factoring help solve equations?
If a product equals zero, one factor must be zero, so you set each factor to zero.
Try it yourself
Polynomial CalculatorAdd, subtract and multiply polynomials and evaluate the result at any x.Quadratic Equation SolverSolve ax² + bx + c = 0 with the discriminant, roots and vertex.
Prime Factorization CalculatorBreak a number into primes and see its divisor count and divisor sum.Keep learning
Polynomials Explained8 min readHow to Solve Quadratic Equations8 min readThe Remainder and Factor Theorems7 min readWritten and reviewed by the SolveCalcPro editorial team. Spot a mistake? Tell us. See our editorial policy.