SPEC A4 • ALGEBRA — FACTORISING
Triple Brackets & Factorising — Common Factor & DOTS
Higher-tier triple bracket expansion, plus the first two factorising tools: pulling out common factors and spotting the Difference of Two Squares pattern.
What You'll Learn
This lesson has two halves: triple bracket expansion (Higher only) and an introduction to factorising — the mechanical reverse of expanding.
Part 1 — Expanding HIGHER
Triple brackets: pair them up, never multiply all three at once
Part 2 — Factorising (F+H)
Common factor: pulling out the highest shared term
Difference of Two Squares (DOTS): a² − b² = (a + b)(a − b)
"Factorise fully" — the keyword that costs students marks
Why it matters
Factorising is the gateway skill to solving quadratics (Lesson 2.04 and 2.21), simplifying algebraic fractions (2.05), and completing the square (2.22). Fluency here pays off everywhere.
Part 1 Triple Brackets HIGHER
Higher papers may ask you to expand three brackets multiplied together — e.g. (x + 1)(x + 2)(x + 3). The strategy is simple:
The Golden Strategy
Expand in pairs — never all three at once.Step 1: Pick any two brackets and multiply them (using FOIL or grid) — you'll get a trinomial (3-term quadratic).
Step 2: Multiply that trinomial by the third bracket — distribute each of the 3 terms through the 2 terms, giving 6 products.
Step 3: Collect like terms.
Why not all at once?
AQA examiner reports flag this as a common error. Students try to "long multiply" across three brackets simultaneously, which creates 8 products with multiple sign opportunities to track — and sign errors compound. Pair up, every time.
01Worked Example — Triple Bracket H
Expand and simplify: (x + 1)(x + 2)(x + 3) [3 marks]
Step 1 — Expand the first two brackets (FOIL)
(x + 1)(x + 2) = x² + 2x + x + 2 = x² + 3x + 2
Step 2 — Multiply the result by (x + 3)
(x² + 3x + 2)(x + 3)
= x²·x + x²·3 + 3x·x + 3x·3 + 2·x + 2·3
= x³ + 3x² + 3x² + 9x + 2x + 6
Step 3 — Collect like terms
= x³ + 6x² + 11x + 6
Mark scheme split (AQA standard)
M1 for the first expansion with at least three of four terms correct. M1dep for full expansion with all six products of trinomial × binomial shown. A1 for the fully simplified cubic. Showing the trinomial stage earns the first method mark even if the final answer is wrong.
02Worked Example — Authentic AQA Pattern H
Expand and simplify: (x − 2)(x + 3)(x − 5) [3 marks]NOV 2024 • P1 • Q19 PATTERN
Method
Pair two brackets (easiest: first two). Expand with FOIL. Then multiply the result by the third bracket, distributing all six products.
Calculate
(x−2)(x+3) = x² + 3x − 2x − 6 = x² + x − 6
(x² + x − 6)(x − 5):
x²·x = x³, x²·(−5) = −5x²
x·x = x², x·(−5) = −5x
(−6)·x = −6x, (−6)·(−5) = +30
Evaluate
x³ + (−5+1)x² + (−5−6)x + 30
= x³ − 4x² − 11x + 30
Sign check
Evaluate at x = 1 in the original: (1−2)(1+3)(1−5) = (−1)(4)(−4) = 16. Evaluate the cubic: 1 − 4 − 11 + 30 = 16. ✓ Tick — a quick check worth 10 seconds in the exam.
Part 2 Factorising — Vocabulary
Factorising is the reverse of expanding. Instead of removing brackets, you put them back in by finding what two (or more) things multiply to give you the original expression.
Factorise
To write as a product. The inverse of expanding.Expand: 3x(2x + 3) → 6x² + 9xFactorise: 6x² + 9x → 3x(2x + 3)
Common Factor
A term (number or variable) that divides every term in an expression. Always look for the highest common factor (HCF) — number part AND letter part.
Difference of Two Squares (DOTS)
The special factorisation pattern:a² − b² ≡ (a + b)(a − b)
Common Factor Method
How to factorise a common factor
Find the HCF of the numbers (e.g. HCF of 6 and 9 is 3).
Find the lowest power of each letter that appears in every term (e.g. in 6x² + 9x, x is in both — lowest power is x¹).
Combine those: that's your common factor.
Pull it out — divide each term of the original by the common factor, leaving the bracket.
Check by expanding — you should get the original back.
Worked example: 6x² + 9x
HCF of 6 and 9 = 3. Lowest power of x in both terms = x. So common factor = 3x .6x² ÷ 3x = 2x and 9x ÷ 3x = 36x² + 9x = 3x(2x + 3)
03Worked Example — Common Factor
Factorise fully: 4a²b − 10ab² [2 marks]
Step 1 — HCF of numbers
HCF(4, 10) = 2
Step 2 — Lowest power of each letter
a², a¹ → lowest = a. b¹, b² → lowest = b.
Step 3 — Combine: common factor = 2ab
4a²b ÷ 2ab = 2a 10ab² ÷ 2ab = 5b
Step 4 — Write the factorised form
= 2ab(2a − 5b)
Step 5 — Check by expanding
2ab × 2a = 4a²b ✓ 2ab × (−5b) = −10ab² ✓
Mark scheme split
B2 for the fully correct factorisation. B1 for partial credit — e.g. pulling out just 2 or just ab instead of the full HCF 2ab, or getting 2ab(2a − 5b) with a sign slip.
Difference of Two Squares (DOTS)
A very specific pattern — recognise it instantly and it factorises in one line.
The DOTS Identity
a² − b² ≡ (a + b)(a − b)Two conditions must both hold:
Both terms are perfect squares
They are separated by a minus sign
Why it works (check by expanding)
(a + b)(a − b) = a² − ab + ab − b² = a² − b²The middle terms cancel — that's what makes DOTS special.
DOTS in Action
The trick is spotting the pattern. First rewrite each term as something-squared, then read off a and b.
x² − 25
= x² − 5²
(x + 5)(x − 5)
x² − 49
= x² − 7²
(x + 7)(x − 7)
9x² − 16= (3x)² − 4²
(3x + 4)(3x − 4)
25x² − 4= (5x)² − 2²
(5x + 2)(5x − 2)
DOTS can have two variables
Nothing says "b" has to be a plain number. Both terms just need to be perfect squares.4x² − 25y² = (2x)² − (5y)² = (2x + 5y)(2x − 5y)
Recognition checklist
Two terms. Minus between them. Both perfect squares. If all three tick, it's DOTS — factorise in one line.
04Worked Example — DOTS
Factorise: 9x² − 25 [2 marks]
Step 1 — Check for DOTS
Two terms? ✓ Minus sign between them? ✓ Both perfect squares? ✓ (9x² = (3x)² and 25 = 5²)
Step 2 — Identify a and b
a = 3x, b = 5
Step 3 — Apply (a + b)(a − b)
9x² − 25 = (3x + 5)(3x − 5)
Check by expanding
(3x + 5)(3x − 5) = 9x² − 15x + 15x − 25 = 9x² − 25 ✓
False DOTS — The Sum-of-Squares Trap
DOTS only works for differences. A sum of two squares does NOT factorise over real numbers — and students lose marks guessing otherwise.
DOTS ✓
x² − 25 = (x + 5)(x − 5)
Minus sign between perfect squares. Valid DOTS.
NOT DOTS ✗
x² + 25 does not factorise
Plus sign — no factorisation over the real numbers. Leave it alone.
DOTS ✓
4x² − 9 = (2x + 3)(2x − 3)
(2x)² − 3². Valid DOTS.
NOT DOTS ✗
4x² + 9 does not factorise
Sum of squares — cannot be factorised at GCSE level.
Common trap
Students see "square plus square" and invent a factorisation like (2x + 3)(2x + 3) — that's wrong. Expanding (2x + 3)(2x + 3) gives 4x² + 12x + 9, not 4x² + 9. If it's a sum of squares, write "does not factorise" and move on.
"Factorise FULLY" — The Keyword
When a question says "factorise fully" (or "completely"), it means: keep going until no factor can be broken down any further.
The Strategy
Factorising fully is usually a two-layer process:
Pull out any common factor first (simplest step).
Check if what remains inside the bracket is itself a DOTS (or a factorisable quadratic, covered in Lesson 2.04).
If yes, factorise again.
WEAK STUDENT ✗2x³ − 8x = 2x(x² − 4)
Stopped after one step. But x² − 4 is still a DOTS! Drops from 3/3 to 2/3.
STRONG STUDENT ✓2x³ − 8x = 2x(x + 2)(x − 2)
Common factor then DOTS. All three factors irreducible — fully factorised.
MCE Worked Example — "Factorise Fully"
Factorise fully:
2x³ − 8x
[3 marks]
Method
Look for a common factor first. Then check if anything inside the bracket is still a DOTS.
Calculate
HCF of 2 and 8 is 2; lowest power of x is x. Common factor = 2x.
2x³ − 8x = 2x(x² − 4)
x² − 4 is a DOTS: x² − 2² = (x + 2)(x − 2).
Evaluate
= 2x(x + 2)(x − 2)
Mark scheme (typical 3-mark layered factorisation)
B3 for the fully-factorised answer 2x(x + 2)(x − 2). B2 for 2x(x² − 4) — common factor correct but DOTS not applied. B1 for just pulling out 2 or identifying the common factor without a correct intermediate expression. Show every step — method is in the working.
Examiner Alerts
Alert 1 — "Fully" missed
Students write: 2x³ − 8x = 2x(x² − 4) — stopped after the common factor.
Write instead: Always check the bracket contents. If there's still a DOTS or a factorisable quadratic inside, factorise again. Fully factorised = 2x(x + 2)(x − 2).
Alert 2 — Incorrect DOTS signs
Students write: x² − 4 = (x − 2)(x − 2) — both brackets have the same sign.
Write instead: DOTS requires opposite signs in the two brackets: (x + 2)(x − 2). (x − 2)(x − 2) is (x − 2)² = x² − 4x + 4 — a completely different expression.
Alert 3 — False DOTS on a sum of squares
Students write: 4x² + 9 = (2x + 3)(2x − 3) — invented a DOTS from a sum.
Write instead: DOTS needs a minus sign between the squares. 4x² + 9 has a plus — it does not factorise over the reals. Leave it and move on.
Common Errors
Students often writeCorrect versionWhy it matters
2x³ − 8x = 2x(x² − 4)= 2x(x + 2)(x − 2)"Fully" missed — DOTS remaining inside
x² − 4 = (x − 2)(x − 2)= (x + 2)(x − 2)DOTS needs opposite signs in the brackets
4x² + 9 = (2x + 3)(2x − 3)"Does not factorise"Sum of squares cannot be factorised
6x² + 9x = 3(2x² + 3x)= 3x(2x + 3)Missed the x as part of the common factor
9x² − 16 = (9x + 16)(9x − 16)= (3x + 4)(3x − 4)a = √(9x²) = 3x, not 9x
Multiplied (x+1)(x+2)(x+3) in one stepPair up: (x+1)(x+2) first, then × (x+3)Triple expansion in one pass = 8 products, prone to sign errors
Practice Question 1
Factorise fully:
18x³ − 8x
[3 marks]
M
Method. Common factor first, then check if the bracket is a DOTS.
C
Calculate.
HCF(18, 8) = 2. Lowest power of x = x. Common factor = 2x.
18x³ − 8x = 2x(9x² − 4).
Check: 9x² − 4 = (3x)² − 2² — a DOTS with a = 3x, b = 2.
So 9x² − 4 = (3x + 2)(3x − 2).
E
Evaluate. Final answer: 2x(3x + 2)(3x − 2).
Practice Question 2 H
Expand and simplify:
(x + 1)(x − 2)(x + 4)
[3 marks]
M
Method. Pair the first two brackets and expand (FOIL). Then multiply the resulting trinomial by the third bracket.
C
Calculate.
(x + 1)(x − 2) = x² − 2x + x − 2 = x² − x − 2.
(x² − x − 2)(x + 4):
x²·x + x²·4 + (−x)·x + (−x)·4 + (−2)·x + (−2)·4
= x³ + 4x² − x² − 4x − 2x − 8
= x³ + (4 − 1)x² + (−4 − 2)x − 8
E
Evaluate. Final answer: x³ + 3x² − 6x − 8.
Lesson Summary
Triple brackets H
Pair up — expand two brackets first, then multiply the result by the third
6 products when multiplying a trinomial by a binomial — show them all
AQA awards M1 / M1dep / A1
Factorising — the two tools from this lesson
TypePatternExample
Common factorHCF (number × lowest power of each letter)6x² + 9x = 3x(2x + 3)
DOTSa² − b² = (a + b)(a − b)9x² − 25 = (3x + 5)(3x − 5)
Top 3 traps to avoid
"Fully" ignored — always check the bracket for a further DOTS or quadratic
Wrong DOTS signs — two brackets must be (a + b)(a − b), not (a − b)(a − b)
False DOTS on a sum — x² + 25 does not factorise
Next lesson
Lesson 2.04 — Factorising Quadratics: the third and most important factorising tool — breaking down x² + bx + c (monic) and ax² + bx + c (non-monic, Higher).
Next: What You'll Learn