Multiplying Brackets (Expansion)
Master the distributive law, FOIL, special products, and expanding with confidence — the foundation of all algebra.
Algebraic Expressions
Table of Contents
Algebraic Expressions: Expansion & Factorisation#
Algebra is the foundation of everything in high school maths. Expanding brackets and factorising are opposite skills — you will use them in equations, functions, trigonometry, and every other topic through to matric.
The Two Core Skills#
| Skill | What it does | Direction |
|---|---|---|
| Expansion | Multiplies out brackets into separate terms | Brackets → Terms |
| Factorisation | Rewrites an expression as a product of factors | Terms → Brackets |
They are inverses of each other: $a(b + c) = ab + ac$ works both ways.
Expansion Methods#
1. Single bracket: Distributive law#
$$a(b + c) = ab + ac$$2. Two binomials: FOIL#
$$(x + 3)(x + 5) = x^2 + 5x + 3x + 15 = x^2 + 8x + 15$$First × First, Outer, Inner, Last × Last.
3. Special products (memorise these!)#
| Pattern | Expansion | Example |
|---|---|---|
| Perfect square | $(a + b)^2 = a^2 + 2ab + b^2$ | $(x + 3)^2 = x^2 + 6x + 9$ |
| Difference of squares | $(a + b)(a - b) = a^2 - b^2$ | $(x + 5)(x - 5) = x^2 - 25$ |
⚠️ THE classic error: $(x + 3)^2 \neq x^2 + 9$. The middle term ($2ab = 6x$) is NEVER zero. Always expand properly.
The Factorisation Toolkit#
Always try these in order:
| Step | Method | Example |
|---|---|---|
| 1. | Common factor (always check first!) | $6x^2 + 9x = 3x(2x + 3)$ |
| 2. | Difference of two squares (DOTS) | $x^2 - 16 = (x + 4)(x - 4)$ |
| 3. | Trinomial ($x^2 + bx + c$) | $x^2 + 7x + 12 = (x + 3)(x + 4)$ |
| 4. | Grouping (4 terms → 2 pairs) | $ax + ay + bx + by = a(x+y) + b(x+y) = (a+b)(x+y)$ |
Trinomial factorising: The method#
For $x^2 + bx + c$: find two numbers that multiply to $c$ and add to $b$.
For $ax^2 + bx + c$ (where $a \neq 1$): find two numbers that multiply to $ac$ and add to $b$, then split the middle term and group.
Deep Dives (click into each)#
- Multiplying Brackets & Special Products — FOIL, perfect squares, difference of squares, expanding cubes
- Factorisation: The Complete Toolkit — common factor, DOTS, trinomials, grouping, sum/difference of cubes, with worked examples
🚨 Common Mistakes#
- Forgetting the middle term: $(x + 3)^2 = x^2 + 6x + 9$, NOT $x^2 + 9$.
- Sign errors in DOTS: $x^2 - 9 = (x+3)(x-3)$, but $x^2 + 9$ cannot be factorised with real numbers.
- Not taking out the common factor first: Always check for a common factor BEFORE trying other methods. $2x^2 + 10x + 12 = 2(x^2 + 5x + 6) = 2(x+2)(x+3)$.
- Trinomial sign errors: For $x^2 - 7x + 12$, you need two numbers that multiply to $+12$ and add to $-7$: that’s $-3$ and $-4$.
🔗 Related Grade 10 topics:
- Equations & Inequalities — factorising is the key skill for solving equations
- Functions — x-intercepts of parabolas require factorising
📌 Where this leads in Grade 11: Quadratic Equations — factorising quadratics is the primary solving method
⏮️ Fundamentals | 🏠 Back to Grade 10 | ⏭️ Exponents
Factorization: The Complete Toolkit
Master every factorization technique — common factor, DOTS, trinomials, grouping, and sum/difference of cubes — with full worked examples.