How to Factorise: A Comprehensive Guide for Beginners and Beyond

Understanding how to factorise algebraic expressions is a fundamental skill in mathematics that opens doors to solving equations, simplifying expressions, and advancing in algebra. Whether you’re a student tackling algebra for the first time or someone looking to refresh their skills, this guide will walk you through the essential techniques of how to factorise different types of polynomials. We’ll cover everything from basic factoring to more advanced methods, complete with examples to help you master this crucial mathematical concept.

Table of Contents

• What is Factoring in Algebra?
• Basic Factoring Techniques
• How to Factorise Quadratic Expressions
• Special Factoring Cases
• Advanced Factoring Techniques
• Common Factoring Mistakes to Avoid
• Practice Problems with Solutions

What is Factoring in Algebra?

Factoring, also known as factorisation, is the process of breaking down an algebraic expression into simpler parts called factors that, when multiplied together, give the original expression. Learning how to factorise is essential because it:

• Simplifies complex expressions
• Helps solve polynomial equations
• Makes calculations easier
• Is fundamental for higher mathematics

The Basic Principle of Factoring

The core idea behind how to factorise is finding what common elements exist in all terms of an expression. This could be:

• Numerical factors (greatest common divisor)
• Variable factors (common letters with lowest exponents)
• Binomial or polynomial factors

Basic Factoring Techniques

Before tackling complex expressions, let’s master the fundamental methods of how to factorise simple algebraic terms.

1. Factoring Out the Greatest Common Factor (GCF)

The first step in how to factorise any expression is to look for the Greatest Common Factor (GCF) of all terms.

Example: Factor 6x² + 9x

1. Identify GCF of coefficients: GCF of 6 and 9 is 3
2. Identify GCF of variables: x is common to both terms
3. The GCF is 3x
4. Factor out 3x: 3x(2x + 3)

2. Factoring by Grouping

For expressions with four terms, grouping can be an effective method for how to factorise.

Example: Factor x³ + 2x² + 3x + 6

1. Group terms: (x³ + 2x²) + (3x + 6)
2. Factor each group: x²(x + 2) + 3(x + 2)
3. Factor out common binomial: (x + 2)(x² + 3)

How to Factorise Quadratic Expressions

Quadratic expressions (ax² + bx + c) are among the most common polynomials you’ll need to factor.

1. Simple Trinomial Factoring (a = 1)

When the coefficient of x² is 1, follow these steps:

1. Find two numbers that multiply to c and add to b
2. Write as (x + m)(x + n) where m and n are those numbers

Example: Factor x² + 5x + 6

Numbers that multiply to 6 and add to 5: 2 and 3

Solution: (x + 2)(x + 3)

2. Complex Trinomial Factoring (a ≠ 1)

For quadratics where a ≠ 1, use the “ac method”:

1. Multiply a and c
2. Find two numbers that multiply to ac and add to b
3. Rewrite the middle term using these numbers
4. Factor by grouping

Example: Factor 2x² + 7x + 3

1. ac = 2 × 3 = 6
2. Numbers: 6 and 1 (6×1=6, 6+1=7)
3. Rewrite: 2x² + 6x + x + 3
4. Group: (2x² + 6x) + (x + 3) = 2x(x + 3) + 1(x + 3)
5. Factor: (x + 3)(2x + 1)

Special Factoring Cases

Recognizing these special forms makes how to factorise much easier.

1. Difference of Squares

a² – b² = (a + b)(a – b)

Example: x² – 9 = (x + 3)(x – 3)

2. Perfect Square Trinomials

a² + 2ab + b² = (a + b)²

a² – 2ab + b² = (a – b)²

Example: x² + 6x + 9 = (x + 3)²

3. Sum and Difference of Cubes

a³ + b³ = (a + b)(a² – ab + b²)

a³ – b³ = (a – b)(a² + ab + b²)

Example: 8x³ – 27 = (2x – 3)(4x² + 6x + 9)

Advanced Factoring Techniques

1. Factoring by Substitution

For complex expressions, substitution can simplify the factoring process.

2. Factoring Completely

Always check if factors can be factored further.

Common Factoring Mistakes to Avoid