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Factorising with the Highest Common Factor

Maths · Y8 Algebra
Year 8
Lesson Guide
Explanation
Factorising is expanding run backwards: it rewrites a SUM as a PRODUCT. Expanding turns 3(x + 4) into 3x + 12; factorising spots that 3x + 12 has a common factor of 3 in both terms and pulls it out the front: 3(x + 4). The glorious part: every factorisation carries a built-in check — expand your answer, and you must land exactly back on the original. Factorise, expand-to-check, done. No other topic self-checks this cleanly.
Your Turn
Factorise 5y + 20, then expand to check. Check: 5(y + 4) → 5y + 20 ✓.
Watch Out
A factorised answer is a PRODUCT — something(bracket). If your answer still has a top-level + between separate pieces, you haven't factorised.
Explanation
Before pulling a factor out, find the biggest one — the HCF. For terms it has two parts: the HCF of the coefficients, times each shared letter at its LOWEST power. For 12x2 and 18x: numbers give 6; the letters share x (lowest power x1, since 18x has just one). HCF = 6x. This is the same HCF you used to simplify ratios in Chapter 3 — letters simply join in, and a letter both terms don't share contributes nothing.
Your Turn
Find the HCF of 8ab and 12b, and of 9m2 and 15m. Check: 4b; 3m.
Watch Out
The letter must be in EVERY term to join the HCF: for 6xy and 9x, only x qualifies (HCF 3x) — y is missing from the second term.
Explanation
The three-step method. Step 1: find the HCF and park it outside a bracket — 12x2 + 18x has HCF 6x, so write 6x( ). Step 2: fill the bracket by DIVIDING each original term by the HCF: 12x2 ÷ 6x = 2x and 18x ÷ 6x = 3, giving 6x(2x + 3). That division is last lesson's term division earning its keep. Step 3: expand to check — 6x × 2x = 12x2 ✓, 6x × 3 = 18x ✓. The bracket holds what remains after the HCF walks out.
Your Turn
Factorise 10a2 + 15a. Check: HCF 5a → 5a(2a + 3); expand to confirm.
Watch Out
Pull out the FULL HCF: 12x2 + 18x = 2(6x2 + 9x) is true but unfinished — the bracket still shares 3x. Fully factorised means the bracket has nothing left in common.
Explanation
When the first term is negative, pull out a NEGATIVE HCF — it keeps the bracket tidy. −6x − 9: take out −3, and each division flips signs: −6x ÷ (−3) = 2x, −9 ÷ (−3) = 3, giving −3(2x + 3). Trickier: −4m + 10 = −2(2m − 5) — watch the +10 become −5 inside. Dividing by a negative flips every sign, which is exactly why the expand-check matters most here: −2 × (−5) = +10 ✓.
Your Turn
Factorise −8y − 12, and −5a + 15. Check: −4(2y + 3); −5(a − 3).
Watch Out
After pulling out a negative, EVERY sign inside flips. Expand to check — sign slips here are near-universal.
Explanation
Multi-letter terms, same machine. 8a2b + 12ab2: the number HCF is 4; both terms have at least one a and one b, so the letter part is ab. HCF = 4ab. Divide each term: 8a2b ÷ 4ab = 2a, and 12ab2 ÷ 4ab = 3b. Answer: 4ab(2a + 3b). Three-term expressions join in too: 6x2 + 9x + 3 = 3(2x2 + 3x + 1) — every term must be divided, and the lonely 3 ÷ 3 leaves a 1, never nothing.
Your Turn
Factorise 10xy + 15x2, and 4p + 8pq + 12p2. Check: 5x(2y + 3x); 4p(1 + 2q + 3p).
Watch Out
When a term IS the HCF, dividing leaves 1, not 0: 4p + 8pq = 4p(1 + 2q). Drop the 1 and expanding won't return the 4p.
Explanation
Why bother factorising? Shortcuts now, power later. Now: 47 × 8 + 53 × 8 = 8(47 + 53) = 8 × 100 = 800, instantly. Later: factorised form is how quadratic equations get SOLVED in Year 9 — a product equalling zero cracks open in a way a sum never does. And the chapter closes its loop: vocabulary → like terms → multiplying and dividing → index laws → expanding → factorising, with expand-and-factorise as perfect inverses. Master the loop and the quiz is yours!
Your Turn
Capstone: factorise 9x2 − 6x, then expand to verify. Check: 3x(3x − 2) → 9x2 − 6x ✓.
Watch Out
Exam wording matters: 'expand' means remove brackets, 'factorise' means create them. Read which direction the question wants.
📌 Key Points — this lesson in a nutshell
  • Factorising rewrites a sum as a product — expanding in reverse: 3x + 12 = 3(x + 4).
  • HCF of terms = HCF of the coefficients × each shared letter at its lowest power.
  • Method: HCF out front, fill the bracket by dividing each term by the HCF.
  • ALWAYS expand your answer to check — factorising self-checks perfectly.
  • First term negative? Pull out a negative HCF and flip every sign inside: −4m + 10 = −2(2m − 5).
  • A term that IS the HCF leaves 1 behind, never nothing: 4p + 8pq = 4p(1 + 2q).
  • Fully factorised = the bracket's terms share no more common factors.
Step 1: Introduction
Step 1 of 6
The two-way street 3(x + 4) 3x + 12 expand factorise Factorising rewrites a SUM as a PRODUCT of factors. 3x + 12 = 3(x + 4): pull the common factor 3 out the front. Instant check: expand your answer — you must land back where you started. Every factorisation carries its own built-in answer check!
HCF of 12x2 and 18x 12x2 = 2·2·3 · x·x 18x = 2·3·3 · x shared: 2 · 3 · x Shortcut: HCF of numbers: 6 lowest power of x: x1 HCF = 6x Rule: HCF of the coefficients × each shared letter at its LOWEST power. Same HCF idea as Chapter 3's ratio simplifying — now with letters.
Factorise 12x2 + 18x Step 1 — HCF: 6x. Write it out front: 6x( ) Step 2 — fill the bracket by dividing each term: 12x2 ÷ 6x = 2x 18x ÷ 6x = 3 = 6x(2x + 3) Step 3 — CHECK by expanding: 6x × 2x = 12x2 ✓ 6x × 3 = 18x ✓ The bracket holds what's LEFT after the HCF walks out.
Factorise −6x − 9 Pull out −3: −3(2x + 3) −6x ÷ −3 = 2x; −9 ÷ −3 = 3 Also true: 3(−2x − 3) but a leading negative in the bracket is untidy — prefer pulling the minus out. When the FIRST term is negative, take out a negative HCF. Dividing by a negative flips every sign inside: −4m + 10 = −2(2m − 5) Expand to check: −2 × 2m = −4m ✓ −2 × (−5) = +10 ✓
Factorise 8a2b + 12ab2 HCF hunt: numbers → 4; a's → a1; b's → b1 HCF = 4ab both terms have at least one a and one b 8a2b ÷ 4ab = 2a 12ab2 ÷ 4ab = 3b term ÷ HCF, using last lesson's division = 4ab(2a + 3b) Three terms work the same way: 6x2 + 9x + 3 = 3(2x2 + 3x + 1).
Chapter 4 complete 🏆 Vocabulary terms, coefficients, constants Like terms collect species; signs travel × and ÷ terms numbers, letters, powers Index laws add / subtract / multiply; a0=1 Expanding a(b+c) = ab + ac Factorising HCF out front, expand to check Factorised form finds mental shortcuts (47×8 + 53×8 = 100×8) and, next year, SOLVES equations. Expanding ⇄ factorising: one loop. Quiz!
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