Factorising Algebraic Expressions Study Notes

Factorising Algebraic Expressions Overview

  • Essential techniques:

    • Factorising using common factors

    • Factorising by grouping in pairs

    • Factorising using the difference of two squares

    • Factorising quadratic trinomials

    • Simplifying algebraic fractions through factorisation

Key Learning Outcomes

  • Mastery in factorising various types of algebraic expressions.

  • Ability to simplify calculations with algebraic fractions by factorising.

Factorising Techniques

  • Common Factors:

    • Identify the highest common factor (HCF) of terms.

    • Example: 5a^2 - 10a = 5a(a - 2).

      • Step-by-step: Find the HCF of 5a^2 and 10a, which is 5a. Divide both terms by the HCF: (5a^2 \div 5a = a) and (10a \div 5a = 2). Place the HCF outside the brackets and the quotients inside.

  • Grouping in Pairs:

    • Rearrange and group terms with common factors.

    • Example: 3x + 3 + mx + m = (x + 1)(3 + m).

      • Step-by-step: Group the terms into two pairs: (3x + 3) + (mx + m). Factor out the HCF from each pair: 3(x + 1) + m(x + 1). Since (x + 1) is now a common factor for the whole expression, factor it out to leave (x + 1)(3 + m).

  • Difference of Two Squares:

    • Formula: a^2 - b^2 = (a - b)(a + b).

    • Example: x^2 - 4 = (x - 2)(x + 2).

      • Step-by-step: Recognize that 4 = 2^2, making the expression x^2 - 2^2. Using the formula where a = x and b = 2, write the factors as (x - 2)(x + 2).

  • Quadratic Trinomials:

    • Identify two numbers that add to the coefficient of x and multiply to the constant term.

    • Example: x^2 + 5x + 6 = (x + 2)(x + 3).

      • Step-by-step: Identify the constant (6) and the coefficient of x (5). Find two numbers that multiply to 6 and add to 5. These numbers are 2 and 3 (2 \times 3 = 6; 2 + 3 = 5). Place these into the brackets: (x + 2)(x + 3).

Further Factorisation Methods

  • Using Cross Method:

    • For cases where the leading coefficient is not one, apply cross multiplication to find factors.

Miscellaneous Types of Factorisation

  • Combine techniques from previous sections to factorise expressions with multiple terms.

Simplifying Algebraic Fractions

  • Factor both numerator and denominator to find common factors to simplify.

Mathematical Definitions

  • Binomial: An expression with two terms.

  • Coefficient: The numeric factor in a term.

  • Expand: Remove grouping symbols by distributing.

  • Factorise: Express an expression as a product of its factors.

  • Trinomial: An expression with three terms, typically involving a quadratic form.