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.