Algebra Study Guide

This is Higher-tier content, so solutions should communicate multi-step reasoning precisely. Expanding brackets uses the distributive law: multiply every term in one bracket by every term in the other. With two binomials this creates four products. Combine like terms only after all products, including their signs, have been written.

For example, (x + 3)(x − 5) = x² − 5x + 3x − 15 = x² − 2x − 15. For (2x − 3)(x + 4), the products are 2x², 8x, −3x and −12, giving 2x² + 5x − 12.

A strong solution states the relevant rule, substitutes or transforms values carefully, shows intermediate reasoning, and checks that the result is reasonable and in the required form. Common errors come from omitted terms, mishandled signs, premature rounding or using a rule that does not match the structure of the problem.

Factorising is the reverse process of expanding brackets. When you expand brackets, you multiply terms together; when you factorise, you are looking for common factors to 'take out' of the terms and place outside a bracket. The goal is to rewrite an expression as a product of its factors. For example, in the expression 6x + 9, both 6x and 9 share a common factor of 3. We can write 6x as 3 * 2x and 9 as 3 * 3. Therefore, 6x + 9 can be factorised as 3(2x + 3). To check your answer, expand the factorised expression: 3 * 2x = 6x and 3 * 3 = 9, which gives 6x + 9. This confirms the factorisation is correct. Factorising simplifies expressions and is a fundamental skill used in solving equations, simplifying fractions, and working with more complex algebraic expressions. Always look for the *highest* common factor to ensure the expression is fully factorised. This process applies to terms with variables and/or numbers.