Factoring Calculator
Factor any polynomial or quadratic expression completely — with GCF, trinomials, difference of squares, and step-by-step solutions.
Try: x^2 + 5x + 6 , 2x^2 + 7x + 3 , x^2 - 16
Factoring Methods — Quick Reference
| Method | Example | Factored Form |
|---|---|---|
| GCF (Greatest Common Factor) | 6x² + 9x | 3x(2x + 3) |
| Factoring Trinomials (a=1) | x² + 5x + 6 | (x + 2)(x + 3) |
| Factoring Trinomials (a≠1) | 2x² + 7x + 3 | (2x + 1)(x + 3) |
| Difference of Squares | x² - 16 | (x - 4)(x + 4) |
| Sum/Difference of Cubes | x³ - 8 | (x - 2)(x² + 2x + 4) |
| Factor by Grouping | x³ + 2x² + 3x + 6 | (x² + 3)(x + 2) |
| Perfect Square Trinomial | x² + 6x + 9 | (x + 3)² |
How to Factor — Step-by-Step Strategy
- 1Factor out the GCF first: Always look for the Greatest Common Factor before anything else. Pull out any common numerical or variable factor from all terms.
- 2Count the terms: 2 terms → try difference of squares or sum/difference of cubes. 3 terms → try trinomial factoring. 4 terms → try grouping.
- 3Check if it's a special pattern: Difference of squares: a²−b² = (a−b)(a+b). Perfect square: a²+2ab+b² = (a+b)². Sum of cubes: a³+b³ = (a+b)(a²−ab+b²).
- 4Factor trinomials: For x² + bx + c: find two numbers that multiply to c and add to b. For ax² + bx + c (a≠1): use the AC method or trial and error.
- 5Check by expanding: Multiply your factors back out (FOIL or distribution) to verify the result equals your original expression.
Frequently Asked Questions
What does it mean to factor a polynomial?
Factoring a polynomial means rewriting it as a product of simpler expressions (factors). For example, x² + 5x + 6 factors into (x + 2)(x + 3). Factoring is the reverse of expanding or multiplying polynomials.
How do I factor a quadratic expression?
To factor a quadratic ax² + bx + c: (1) Find two numbers that multiply to a×c and add to b. (2) Rewrite the middle term using those numbers. (3) Factor by grouping. Example: x² + 5x + 6 → find 2 and 3 (2×3=6, 2+3=5) → (x+2)(x+3).
What is the Greatest Common Factor (GCF) method?
The GCF method finds the largest expression that divides evenly into every term. Factor it out using the distributive property. Example: 6x² + 9x → GCF is 3x → 3x(2x + 3). Always check for GCF first before applying other methods.
What is the difference of squares formula?
The difference of squares formula states that a² - b² = (a - b)(a + b). Example: x² - 25 = (x - 5)(x + 5). This pattern only works for subtraction (difference), not addition. x² + 25 cannot be factored over real numbers.
How do I factor by grouping?
Factor by grouping works when a polynomial has 4 terms: (1) Split into two pairs, (2) Factor out the GCF from each pair, (3) Factor out the common binomial. Example: x³ + 2x² + 3x + 6 → x²(x+2) + 3(x+2) → (x²+3)(x+2).