Factorising Expressions Quizzes

Factorising Expressions Quiz 1

Difficulty: Foundation

Curriculum: GCSE

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Factorising Expressions Quiz 3

Difficulty: Higher

Curriculum: GCSE

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Factorising Expressions Quiz 2

Difficulty: Foundation

Curriculum: GCSE

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Factorising Expressions Quiz 3

Difficulty: Higher

Curriculum: GCSE

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Factorising Expressions featured image

Visual overview of Factorising Expressions.

Introduction

Factorising expressions is like uncovering the hidden structure within algebra. It means rewriting an expression as a product of simpler factors that multiply to give the original form. This reverses the process of expansion and reveals patterns that make equations easier to solve. Mastering factorisation links multiplication and addition, turning complex expressions into manageable parts.

Example: \(6x+9=3(2x+3)\).

Core Concepts

What is Factorising?

Factorising means expressing a sum or difference as a product. It is the opposite of expanding brackets.

  • Expanded form → \(6x+9\)
  • Factorised form → \(3(2x+3)\)
Think of it as “putting brackets back in.”

Common Factors (Single Bracket Factorisation)

Find the greatest common factor (GCF) of all terms and factor it out.

  • \(8x+12=4(2x+3)\)
  • \(15a+20b=5(3a+4b)\)
  • \(9x^2+6x=3x(3x+2)\)
Rule: Divide each term by the GCF and write what remains inside brackets.

Factorising with Negative Signs

When the first term is negative, it is often easier to take out a negative factor.

  • \(-6x-9=-3(2x+3)\)
  • \(-10a+20b=-10(a-2b)\)
Tip: Factor out the negative to make the first bracket term positive.

Difference of Two Squares

Pattern: \(a^2-b^2=(a+b)(a-b)\)

  • \(x^2-9=(x+3)(x-3)\)
  • \(4x^2-25=(2x+5)(2x-5)\)
  • \(9y^2-16=(3y+4)(3y-4)\)
Check: both terms must be perfect squares separated by a minus sign only.

Factorising Quadratic Trinomials

A quadratic expression has the form \(ax^2+bx+c\). We look for two numbers that multiply to \(a\times c\) and add to \(b\).

Example

Simplify \(x^2+7x+12\).

  • Find two numbers that multiply to \(12\) and add to \(7\): \(3\) and \(4\).
  • Write: \((x+3)(x+4)\).

Quadratics with a Coefficient of x² ≠ 1

For \(ax^2+bx+c\) where \(a>1\), use grouping or trial pairs of factors.

Example

Simplify \(2x^2+5x+2\).

  • Product \(a\times c=4\); numbers adding to \(5\): \(4\) and \(1\).
  • Rewrite middle term: \(2x^2+4x+x+2\)
  • Group: \((2x^2+4x)+(x+2)\)
  • Factor each: \(2x(x+2)+1(x+2)\)
  • Final: \((2x+1)(x+2)\)

Factorising with Common Brackets (Grouping)

When terms share a common bracket, take that bracket out as a factor.

  • \(3x(x+4)+2(x+4)=(3x+2)(x+4)\)

Special Cases and Perfect Squares

When a quadratic is a perfect square trinomial:

  • \(x^2+6x+9=(x+3)^2\)
  • \(4x^2-12x+9=(2x-3)^2\)
Spot the pattern: the first and last terms are squares, and the middle term is twice their product.

Worked Examples

Example 1 (Foundation): Common Factor

Factorise \(9x+12\).

  • GCF \(=3\)
  • \(3(3x+4)\)

Example 2 (Foundation): With Negative Sign

Factorise \(-15x-20\).

  • GCF \(=-5\)
  • \(-5(3x+4)\)

Example 3 (Higher): Difference of Squares

Factorise \(x^2-49\).

  • \((x+7)(x-7)\)

Example 4 (Higher): Quadratic with 1x²

Factorise \(x^2+8x+15\).

  • Pairs for 15 that sum to 8 → 3 & 5
  • \((x+3)(x+5)\)

Example 5 (Higher): Quadratic with 2x²

Factorise \(2x^2+7x+3\).

  • Product \(a\times c=6\); pair \(6,1\)
  • \(2x^2+6x+x+3=(2x^2+6x)+(x+3)\)
  • \(2x(x+3)+1(x+3)=(2x+1)(x+3)\)

Example 6 (Higher): Grouping with Common Bracket

Factorise \(4x(x-2)+5(x-2)\).

  • \((x-2)(4x+5)\)

Common Mistakes

  • Not taking out the greatest common factor.
  • Forgetting to divide every term by the factor.
  • Sign errors when the first term is negative.
  • Using incorrect pairs of numbers in quadratics.
  • Trying to apply the difference-of-squares rule with a “+” sign.
Check: Multiply your factors back out—if you get the original expression, the factorisation is correct.

Applications

  • Solving equations by setting each bracket equal to zero.
  • Rewriting formulas for simplification or cancellation.
  • Understanding the structure of quadratic graphs.
  • Simplifying fractions with algebraic numerators and denominators.

Strategies & Tips

  • Always factor out any common number or variable first.
  • Memorise special patterns: \(a^2-b^2\) and perfect squares.
  • For quadratics, test factors of \(a\times c\) that sum to \(b\).
  • Check your result by expanding it back out.
  • Practise both simple and complex cases until pattern spotting feels natural.

Summary / Call-to-Action

Factorising reveals the hidden structure of algebra. By taking out common factors, recognising squares, and decomposing quadratics, you can simplify, solve, and understand expressions at a deeper level. These methods turn complex algebra into step-by-step reasoning.

  • Practise each method—common factor, difference of squares, quadratic.
  • Expand back after each attempt to verify accuracy.
  • Apply factorisation when solving equations or simplifying fractions.
  • Challenge yourself with mixed questions across all patterns.