How to Factorise: A Comprehensive Guide
Factorising, also known as factoring, is a fundamental concept in algebra. It involves breaking down a mathematical expression into simpler components that, when multiplied together, give the original expression. This skill is crucial for solving equations, simplifying expressions, and understanding various mathematical concepts. This guide will walk you through different methods of factorising, focusing on common scenarios.
Understanding the Basics
Before diving into techniques, let's clarify what factorising means. Consider the number 12. Its factors are 1, 2, 3, 4, 6, and 12 because these numbers multiply together to give 12 (e.g., 2 x 6 = 12, 3 x 4 = 12). Factorising algebraic expressions works similarly; we aim to find expressions that, when multiplied, give the original expression.
Common Factorising Techniques
Several methods exist for factorising, depending on the structure of the expression. Here are some of the most frequently used:
1. Finding the Greatest Common Factor (GCF)
This is the simplest method. It involves identifying the greatest factor common to all terms in the expression and factoring it out.
Example: Factorise 3x + 6y
- Identify the GCF: The GCF of 3x and 6y is 3.
- Factor out the GCF: 3(x + 2y)
2. Factorising Quadratic Expressions (ax² + bx + c)
Quadratic expressions are trinomials (three terms) with the highest power of the variable being 2. Several techniques can factorise these:
- Simple Factoring: If the coefficient of x² (a) is 1, look for two numbers that add up to 'b' and multiply to 'c'.
Example: Factorise x² + 5x + 6
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Find the numbers: The numbers 2 and 3 add up to 5 and multiply to 6.
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Factorise: (x + 2)(x + 3)
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Factoring with a Leading Coefficient (a ≠ 1): When 'a' is not 1, the process is slightly more complex. You can use the AC method, grouping, or trial and error. The AC method involves multiplying 'a' and 'c', finding two numbers that add to 'b' and multiply to 'ac', and then using grouping to factorise.
Example: Factorise 2x² + 7x + 3
- AC method: a x c = 6. Find two numbers that add to 7 and multiply to 6 (6 and 1).
- Rewrite: 2x² + 6x + x + 3
- Group: 2x(x + 3) + 1(x + 3)
- Factor: (2x + 1)(x + 3)
3. Difference of Squares
This technique applies to expressions in the form a² - b². It factors into (a + b)(a - b).
Example: Factorise x² - 9
- Identify a and b: a = x, b = 3
- Factorise: (x + 3)(x - 3)
4. Sum and Difference of Cubes
These factorisations apply to expressions of the form a³ + b³ and a³ - b³. Their respective factorisations are:
- a³ + b³ = (a + b)(a² - ab + b²)
- a³ - b³ = (a - b)(a² + ab + b²)
Tips for Successful Factorising
- Always look for a GCF first. This simplifies the expression and makes subsequent factorisation easier.
- Check your answer. Expand the factorised expression to ensure it matches the original expression.
- Practice regularly. Factorising becomes easier with consistent practice. Work through various examples and gradually increase the complexity of the expressions.
- Utilize online resources. Many websites and videos offer step-by-step instructions and examples to help you master factorising techniques.
By understanding and applying these methods, you'll be well-equipped to tackle a wide range of factorisation problems. Remember, consistent practice is key to mastering this essential algebraic skill.
