Expanding brackets — the mechanism that unlocks every equation to come.
Section 01
The Core Law
The distributive property states that a factor multiplying a sum (or difference) inside brackets is applied to every term inside. The bracket is expanded; the factor is distributed.
a(b + c) = ab + ac
a(b − c) = ab − ac
3(x + 5)
↙ ↘
3×x + 3×5
= 3x + 15
Distributing is like handing out identical instructions to a group: if a coach tells each player to do 3 sets of push-ups and 3 sets of sprints, it is the same as saying 3 × (push-ups + sprints). Every member of the group gets the full instruction.
Key Point — Every Term Gets Multiplied
The factor outside the brackets multiplies every single term inside — not just the first. This is the most common place for errors: students multiply the first term and forget the rest.
✗ Wrong:4(x + 3) = 4x + 3
✓ Correct:4(x + 3) = 4x + 12
Section 02
Expanding with Positive Factors
Worked Example A — Expand 5(2x − 3)
1
Multiply 5 by each term: 5 × 2x and 5 × (−3)
2
10x and −15
3
Result: 10x − 15
Worked Example B — Expand 3x(4x² − 2x + 1)
1
3x × 4x² = 12x³
2
3x × (−2x) = −6x²
3
3x × 1 = 3x
4
Result: 12x³ − 6x² + 3x
Multiplying Powers — Quick Reminder
When multiplying terms with the same base, add the exponents:
x × x² = x^(1+2) = x³
3x × 4x² = (3×4)(x^1 × x^2) = 12x³
Section 03
Expanding with Negative Factors
When the factor outside the brackets is negative, every sign inside flips. This is where most errors occur.
−a(b + c) = −ab − ac −a(b − c) = −ab + ac
Worked Example C — Expand −4(3x − 7)
1
−4 × 3x = −12x
2
−4 × (−7) = +28
3
Result: −12x + 28
Distributing a Bare Negative Sign
When only a minus sign precedes the brackets, the factor is −1. Every term inside changes sign:
−(x − 5) = −1(x − 5) = −x + 5
Students who drop the negative and write x − 5 unchanged make the single most common expansion error in all of algebra.
Section 04
Expand and Simplify
The real power of the distributive property emerges when you combine it with collecting like terms. The standard task: expand all brackets, then collect like terms.
Worked Example D — Expand and simplify 3(2x + 1) + 4(x − 5)
1
Expand first bracket: 3(2x + 1) = 6x + 3
2
Expand second bracket: 4(x − 5) = 4x − 20
3
Combine: 6x + 3 + 4x − 20
4
Collect: (6x + 4x) + (3 − 20)
5
Result: 10x − 17
Worked Example E — Expand and simplify 5(a − 2b) − 2(3a + b)
1
5(a − 2b) = 5a − 10b
2
−2(3a + b) = −6a − 2b
3
Combine: 5a − 10b − 6a − 2b
4
a-terms: 5a − 6a = −a. b-terms: −10b − 2b = −12b
5
Result: −a − 12b
Section 05
Factorising — The Reverse Process
Factorising is the inverse of expanding. Instead of removing brackets, you introduce them by identifying a common factor across all terms and pulling it out.
ab + ac = a(b + c)
Worked Example F — Factorise 12x² − 8x
1
Find the GCF of 12 and 8: GCF = 4. Common variable: x. So the HCF of both terms is 4x.
2
Divide each term: 12x² ÷ 4x = 3x and 8x ÷ 4x = 2
3
Write as product: 4x(3x − 2)
4
Verify by expanding: 4x(3x − 2) = 12x² − 8x ✓
Verification Rule
Always verify a factorisation by re-expanding. If expansion returns the original expression, the factorisation is correct. This habit catches errors instantly.
Section 06
The Distributive Property in Arithmetic
The distributive property also explains a mental arithmetic technique you may already use intuitively:
Mental Arithmetic — Calculate 7 × 98
1
Rewrite: 7 × (100 − 2)
2
Distribute: 7 × 100 − 7 × 2 = 700 − 14
3
Result: 686
Every time you mentally split a difficult multiplication into two easier ones, you are applying the distributive property. Algebra simply names and formalises a calculation technique the mind already uses.
7. A square room has side length (x + 4) metres. A rectangular alcove of dimensions 2 × (x − 1) is removed from one corner. Write a simplified expression for the remaining floor area.
Square area = (x + 4)² — note: this requires lesson 13 (substitution) to fully expand. For now: Square area = (x+4)(x+4). Alcove area = 2(x − 1) = 2x − 2. Remaining = (x+4)(x+4) − (2x − 2). At this stage, accept: (x+4)² − 2(x−1) as the simplified expression, noting (x+4)² will be expanded in a later lesson.