Factoring
Factoring (or factorisation) in algebra is the process of rewriting an algebraic expression as a product of simpler expressions called factors, reversing the process of expansion. For example, x² − 5x + 6 can be factored as (x − 2)(x − 3). Factoring is essential for solving polynomial equations, simplifying rational expressions, and finding roots, and it is a core skill that underpins much of higher mathematics.
Worked Example
Problem
Factor completely: 3x² − 12x − 36.
Solution
Step 1: Factor out the greatest common factor (GCF). GCF of 3, 12, 36 is 3. 3(x² − 4x − 12) Step 2: Factor the trinomial x² − 4x − 12. Find two numbers that multiply to −12 and add to −4: (−6) × 2 = −12 and (−6) + 2 = −4 ✓ Step 3: Write as product of binomials. 3(x − 6)(x + 2) Step 4: Verify by expanding. 3(x² + 2x − 6x − 12) = 3(x² − 4x − 12) = 3x² − 12x − 36 ✓
Answer
3(x − 6)(x + 2)
Common Factoring Patterns and Formulas
| Pattern Name | Factored Form | Expanded Form | Example |
|---|---|---|---|
| Common factor | a(b + c) | ab + ac | 4x² + 8x = 4x(x + 2) |
| Difference of squares | (a + b)(a − b) | a² − b² | x² − 9 = (x+3)(x−3) |
| Perfect square trinomial (+) | (a + b)² | a² + 2ab + b² | x² + 6x + 9 = (x+3)² |
| Perfect square trinomial (−) | (a − b)² | a² − 2ab + b² | x² − 4x + 4 = (x−2)² |
| Sum of cubes | (a + b)(a² − ab + b²) | a³ + b³ | x³ + 8 = (x+2)(x²−2x+4) |
| Difference of cubes | (a − b)(a² + ab + b²) | a³ − b³ | x³ − 27 = (x−3)(x²+3x+9) |
Interactive Tools
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Related Terms
Polynomial
A polynomial is an algebraic expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, where variables have non-negative integer exponents. The general form is aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where the highest exponent n is called the degree. Polynomials are used extensively in calculus, numerical analysis, and computer science for approximating functions and solving complex problems.
Quadratic Equation
A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2, written in standard form as ax² + bx + c = 0 where a ≠ 0. Its graph is a parabola, and it can have two, one, or no real solutions depending on the value of the discriminant (b² − 4ac). Quadratic equations model projectile motion, area problems, and many optimisation scenarios in physics and engineering.
Quadratic Formula
The quadratic formula is an algebraic formula that gives the solutions (roots) of any quadratic equation ax² + bx + c = 0 directly in terms of its coefficients a, b, and c. It is derived by completing the square on the general quadratic and is the most reliable method for solving quadratics, working even when factoring over integers is impossible. The formula also reveals the nature of the roots through the discriminant b² − 4ac.
From Latin "factor" meaning "one who makes or does", derived from "facere" (to make or do). In mathematics, "factor" was adopted to mean a number or expression that divides another exactly. The algebraic use of factoring developed alongside the general theory of polynomial equations in the 17th–18th centuries.