Algebra Study Guide

Factorisation is the process of writing an algebraic expression as a product of its factors. It's essentially the reverse of expanding brackets. When you expand an expression like 3(x + 2), you multiply 3 by x and 3 by 2 to get 3x + 6. Factorising 3x + 6 means finding the common factors in both terms and 'taking them out' to write the expression in the form 3(x + 2).

The first step in factorising is always to look for the greatest common factor (GCF) of all the terms in the expression. This can be a number, a variable, or a combination of both. For example, in the expression 4x + 8, the GCF of 4x and 8 is 4. We can write 4x as 4 * x and 8 as 4 * 2. So, we 'take out' the 4, and what's left goes inside the bracket: 4(x + 2).

If the expression contains variables, you also look for common variables. For instance, in 3x² + 6x, the numerical common factor for 3 and 6 is 3. The common variable factor for x² and x is x. So, the GCF is 3x. We divide 3x² by 3x to get x, and we divide 6x by 3x to get 2. Therefore, 3x² + 6x factorises to 3x(x + 2).

It's crucial to check your answer by expanding your factorised expression. If you get back to the original expression, you've factorised correctly. Factorisation is a fundamental skill in algebra, used in solving equations, simplifying expressions, and working with fractions.

Factorising is the reverse of expanding brackets. Instead of multiplying expressions out, we are putting them back into brackets. For higher-level factorising, you will encounter more complex forms than simple common factor or basic quadratic factorisation. Key higher techniques include factorising quadratic expressions where the coefficient of x² is not 1 (ax² + bx + c, where a ≠ 1), the difference of two squares (DOTS), and factorising algebraic fractions. For quadratic expressions ax² + bx + c, when a ≠ 1, one common method is to multiply 'a' and 'c', find two numbers that multiply to 'ac' and add to 'b', then rewrite the middle term 'bx' using these two numbers. Finally, factorise by grouping. For example, to factorise 2x² + 7x + 3, multiply 2 by 3 to get 6. Find two numbers that multiply to 6 and add to 7 (which are 1 and 6). Rewrite the expression as 2x² + x + 6x + 3. Then factorise by grouping: x(2x + 1) + 3(2x + 1), which gives (2x + 1)(x + 3). The difference of two squares states that a² - b² = (a - b)(a + b). This is crucial for simplifying expressions and solving equations. Recognising higher common factors, which may themselves be algebraic expressions or have multiple terms, is also vital. For example, factorising 3x(x - 2) + 5(x - 2) involves recognising (x - 2) as a common factor, leading to (x - 2)(3x + 5).