Mixed Exercise – Factorising

Mixed Exercise – Factorising

Introduction

  • The process of factorising involves rewriting an algebraic expression as the product of its factors. These factors are the numbers, variables, expressions, or equations that you multiply together to get the original expression.

  • Factorising can help simplify complex expressions, solve equations and understand the relationships between different algebraic quantities.

Factorising Monic Quadratics

  • Monic quadratics are quadratic expressions with the leading coefficient (coefficient of the squared term) of 1.

  • Factorise a monic quadratic by identifying two numbers that both multiply to give the constant term, and add or subtract to give the coefficient of the linear term.

  • For example, to factorise the monic quadratic x^2 + 5x + 6, note that 2 and 3 multiply to give 6 and add to give 5, so this factorises to (x + 2)(x + 3).

Factorising Non-Monic Quadratics

  • Non-monic quadratics are quadratic expressions where the leading coefficient is not 1.

  • To factorise non-monic quadratics, look for numbers that multiply to give the product of the leading coefficient and the constant term and add or subtract to give the coefficient of the linear term.

  • For example, Look for two numbers that multiply to 10 and add to 7 for the quadratic 2x^2 + 7x + 5, which is (2x + 5)(x + 1).

Factorising by Grouping

  • Factorising by grouping is a technique that can be used when you have four terms in the expression.

  • The process involves grouping pairs of terms together and factoring out a common factor from each group.

  • For example, consider the expression 4x^3 - 2x^2 + 6x - 3. This can be grouped and factorised as 2x^2(2x - 1) + 3(2x - 1), which gives a final factorisation of (2x - 1)(2x^2 + 3).

Factorising Difference of Squares

  • Difference of squares is a special case that can be recognised by the pattern a^2 - b^2. This pattern can be factorised as (a - b)(a + b).

  • For example, the expression x^2 - 9 is a difference of squares, and can be factorised to (x - 3)(x + 3).

Key Points

  • The techniques for factorising involve identifying patterns and pairs of numbers, so practice and familiarity with multiplication and number pairs is important for mastering these skills.

  • The purpose of factorising is to simplify expressions or set up equations to be solved, so always consider how your factorised form can be used in the next step of your algebraic process.