Factorising Expressions Quizzes
Introduction
Factorising expressions is a key algebra skill in GCSE Maths. It involves rewriting expressions as a product of factors. Factorising is the reverse process of expanding brackets and is essential for simplifying expressions, solving equations, and manipulating algebraic formulas. Mastery of factorising helps students solve problems efficiently and is widely used in quadratic equations, linear expressions, and real-life applications.
For example, the expression $$6x + 9$$ can be factorised as $$3(2x + 3)$$. Recognising factors and factorising expressions correctly allows students to work with algebraic structures more effectively.
Core Concepts
Greatest Common Factor (GCF)
The first step in factorising an expression is to identify the greatest common factor of all terms:
- Example: $$12x + 18$$ → GCF = 6 → Factorised: $$6(2x + 3)$$
- Example: $$8a + 12b$$ → GCF = 4 → Factorised: $$4(2a + 3b)$$
Factorising Single Brackets
Take out the GCF from all terms:
Formula: $$ax + ay = a(x + y)$$
Examples:
- $$5x + 10 = 5(x + 2)$$
- $$-3y + 6 = -3(y - 2)$$
- $$2x^2 + 6x = 2x(x + 3)$$
Factorising Double Brackets (Quadratic Expressions)
Quadratic expressions of the form $$ax^2 + bx + c$$ can be factorised by finding two numbers that multiply to $$a \times c$$ and add to $$b$$:
Example:
Factorise $$x^2 + 7x + 12$$
- Numbers that multiply to 12 and add to 7: 3 and 4
- $$x^2 + 3x + 4x + 12$$ → Group terms: $$(x^2 + 3x) + (4x + 12)$$
- Factor each group: $$x(x + 3) + 4(x + 3)$$
- Factor common bracket: $$(x + 4)(x + 3)$$
Factorising by Grouping
Used when an expression has four or more terms:
Example:
Factorise $$ax + ay + bx + by$$
- Group terms: $$(ax + ay) + (bx + by)$$
- Factor each group: $$a(x + y) + b(x + y)$$
- Factor common bracket: $$(a + b)(x + y)$$
Difference of Squares
Formula: $$(a + b)(a - b) = a^2 - b^2$$
Example:
- $$x^2 - 9 = (x + 3)(x - 3)$$
- $$4x^2 - 25 = (2x + 5)(2x - 5)$$
Perfect Square Trinomial
Formula: $$(a + b)^2 = a^2 + 2ab + b^2$$
Factorising: reverse the process
- $$x^2 + 6x + 9 = (x + 3)^2$$
- $$4x^2 - 12x + 9 = (2x - 3)^2$$
Factorising with Negative Signs
Pay attention to negative signs:
- $$-6x - 9 = -3(2x + 3)$$
- $$-x^2 + 5x = -x(x - 5)$$
Factorising Expressions with Indices
Factorise terms with powers of the same variable:
- $$x^3 + x^2 = x^2(x + 1)$$
- $$2x^4 + 6x^3 = 2x^3(x + 3)$$
Substituting Values after Factorising
Factorising first can make substitution easier:
Example:
- Expression: $$6x + 9$$, x = 2
- Factorise: $$3(2x + 3)$$
- Substitute x = 2: $$3(2×2 + 3) = 3(4 + 3) = 3 × 7 = 21$$
Worked Examples
Example 1 (Foundation): Single bracket
Simplify: $$12x + 18$$
- GCF = 6
- Factor out: $$6(2x + 3)$$
Example 2 (Foundation): Factorising with negative
Simplify: $$-8y + 12$$
- GCF = -4
- Factor: $$-4(2y - 3)$$
Example 3 (Higher): Quadratic trinomial
Factorise: $$x^2 + 5x + 6$$
- Numbers that multiply to 6 and add to 5: 2 and 3
- Split middle term: $$x^2 + 2x + 3x + 6$$
- Group: $$(x^2 + 2x) + (3x + 6)$$
- Factor each group: $$x(x + 2) + 3(x + 2)$$
- Common bracket: $$(x + 2)(x + 3)$$
Example 4 (Higher): Difference of squares
Factorise: $$x^2 - 16$$
- $$x^2 - 4^2 = (x + 4)(x - 4)$$
Example 5 (Higher): Perfect square trinomial
Factorise: $$x^2 + 10x + 25$$
- Recognise: $$25 = 5^2, 2ab = 10x → a = x, b = 5$$
- Factor: $$(x + 5)^2$$
Example 6 (Higher): Factorising by grouping
Factorise: $$ax + ay + bx + by$$
- Group: $$(ax + ay) + (bx + by)$$
- Factor each group: $$a(x + y) + b(x + y)$$
- Common bracket: $$(a + b)(x + y)$$
Example 7 (Higher): Factorising with indices
Simplify: $$x^3 + x^2$$
- Factor x^2: $$x^2(x + 1)$$
Example 8 (Real-life): Rectangular area
Length = $$3x + 6$$, Width = 2 → Factorise:
- Factor 3: $$3(x + 2)$$
- Area = Width × Length = 2 × 3(x + 2) = 6(x + 2)$$
Common Mistakes
- Forgetting to factor out the GCF
- Not handling negative signs correctly
- Mixing up factorising vs expanding
- Incorrect factorisation of quadratics
- Ignoring powers when factorising terms with indices
Tips to avoid errors:
- Identify GCF before attempting other factorisation
- Check signs carefully
- Use grouping for four-term expressions
- Practice difference of squares and perfect square trinomials
- Substitute values to verify factorisation
Applications
- Solving linear and quadratic equations
- Algebraic simplification for further operations
- Geometry: calculating area and perimeter of rectangles or squares
- Problem-solving in finance, physics, and real-life contexts
- Preparing expressions for expansion, substitution, and manipulation
Strategies & Tips
- Always start with GCF
- Check for special products: difference of squares or perfect square trinomials
- Use grouping method when four or more terms
- Factor systematically and verify by expanding
- Practice both numeric and algebraic expressions
Summary / Call-to-Action
Factorising expressions is a core algebra skill. By mastering single bracket factorisation, quadratics, grouping, difference of squares, perfect squares, negative signs, and indices, students can simplify and solve problems efficiently. Regular practice ensures accuracy and confidence for GCSE Maths.
Next Steps:
- Attempt factorising quizzes to reinforce learning
- Practice multi-step factorisation problems
- Apply factorising to solve equations and real-life applications
- Challenge yourself with higher-level quadratic and four-term expressions
Consistent practice will make factorising expressions intuitive and error-free.