KS3 · Maths

Factorising expressions (single bracket)

6y + 12 can be written as 2(3y + 6), 3(2y + 4) or 6(y + 2). All correct — only one is finished.

Step 1 · What do they share?

What do 12ab and −20ac have in common?

Every factorisation starts with one question: what do the terms share? Each term has been broken into prime factors and letters — 12ab = 2 × 2 × 3 × a × b and −20ac = −1 × 2 × 2 × 5 × a × c. Put each factor in the circle of the term it comes from, or in the overlap if BOTH terms have it.

  • A Factors of 12ab
  • B Factors of −20ac
  1. 2
  2. 2
  3. 3
  4. 5
  5. −1
  6. a
  7. b
  8. c

Step 2 · Put it outside

Run the area model backwards

Factorising is the area model in reverse. With numbers: 12 + 20 = 4 × 3 + 4 × 5 = 4(3 + 5) — four lots of 3 and four lots of 5 is four lots of (3 + 5). Now do the same with 12ab − 20ac. The two terms are already inside the grid and the width is the HCF you just found, 4a. Find each length by dividing, then write the factorised answer.

Grid: each cell is its row times its column
4a12ab−20ac

Type x² as x^2 if you cannot type ². Spaces do not matter.

Why the HIGHEST?

Terms can share more than one factor

A factor divides a term exactly — nothing left over. For each term on the left, tick whether it is a factor of 6c and whether it is a factor of 15c. Try splitting 6c and 15c into factor pairs first.

1
2
3
5
c
3c
5c

Step 3 · Is it finished?

Fully factorised — or not yet?

Sort each factorisation: is it fully factorised, or factorised but not fully?

Still to sort

Fully factorised (0)

The HCF is outside, so the terms left inside share no common factor.

Where the line is: Look inside the bracket. If the terms there share nothing but 1, the highest common factor must already be outside.

Factorised, but not fully (0)

Correct — but the terms inside still share a factor.

Where the line is: Still a correct factorisation, and sometimes a partly factorised form is useful. But if you're asked to factorise fully, something shared is still hiding in the bracket.

9 of 9 still to sort.

Every answer here is correct. Expand any of them and you get the original back. The question is whether it's finished.

Spot the slip

Where does this answer stop too soon?

Factorise fully: 8x + 40

A student's working — which line goes wrong?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Expanding, run backwards: find what every term shares and put it outside.

What you need to know

  • A factor of a term divides it exactly. Algebraic terms have factors just like numbers do: 2 and y are both factors of 2y, and the factors of 7ab are 1, 7, a, b, 7a, 7b, ab and 7ab.
  • List factors in pairs so none go missing, and never forget 1 and the term itself. Letters count too: x² is x × x, so x is a factor of 15x².
  • The highest common factor (HCF) of some terms is the biggest factor that EVERY term shares. Write each term as prime factors and letters, put the shared ones in the overlap of a Venn diagram, and multiply only what is in the overlap.
  • To factorise, put the HCF outside the bracket and divide each term by it to find what goes inside. If a term is the HCF itself, dividing leaves 1, not 0: 5x + 10xy = 5x(1 + 2y).
  • An expression is fully factorised when the HCF is outside, so the terms left inside the bracket share no common factor. A factorisation that isn't full is still correct — just not finished.

The big picture

Factorising means rewriting an expression as a product: one term outside a bracket, multiplied by everything inside it. It is expanding brackets run backwards. First find the highest common factor (HCF) of the terms — break each term into prime factors and letters, and multiply only the factors they share. Put the HCF outside the bracket, divide each term by it to find what goes inside, then expand your answer to check you get back where you started.

Key points

1Factorising is expanding in reverse: the distributive law run backwards.
2HCF = multiply the overlap ONLY — never every factor in the diagram.
3A minus sign doesn't change the HCF: −20ac is −1 × 20ac, and the −1 stays outside the overlap.
4Divide EVERY term by the HCF, and keep each term's sign inside the bracket.
5Always check by expanding: you must get back exactly what you started with.
6Correct isn't the same as finished — look inside the bracket for anything still shared.

Worked example

Problem

Factorise fully: 8y² − 12y

⚠ Watch out

Taking out a common factor that isn't the highest one. 12y + 18 = 3(4y + 6) is correct, but 4y and 6 still share 2 — so it isn't fully factorised. The HCF is 6, giving 6(2y + 3).

🧠

Memory hook

Outside: what they ALL share. Inside: what's left of each. Then peek inside the bracket — if anything is still shared, you're not done.

✓

Check yourself

Expand 2a(3 + 6b) to see what it came from. Is it fully factorised? If not, finish the job — and watch what's left when a term IS the HCF.

Flashcards

(15)
What is a factor of a term?
A term that divides it exactly, with nothing left over. 2 and y are both factors of 2y, because 2 × y = 2y.
List all the factors of 7ab.
1, 7, a, b, 7a, 7b, ab and 7ab — eight factors.
Why is x a factor of 15x²?
Because x² = x × x, so 15x² = x × 15x. x divides it exactly.
What are the factor pairs of 6a?
1 and 6a, 2 and 3a, 3 and 2a, 6 and a.
What does 'highest common factor' (HCF) mean?
The biggest factor that every term shares. Every other common factor divides into it.
How do you find the HCF of two terms with a Venn diagram?
Write each term as prime factors and letters, put the ones both terms have in the overlap, then multiply only what's in the overlap.
Find the HCF of 10abc and 15ab.
5ab. 10abc = 2 × 5 × a × b × c and 15ab = 3 × 5 × a × b, so the overlap holds 5, a and b.
What is the HCF of 2a, 6ab and 9ab?
a — not 18ab. Only a is in all three terms: 2a has no 3 or b, and 9ab has no 2.
Does a minus sign change the HCF? Try 12a²b² and −20a³c.
No. The −1 belongs to one term only, so it stays out of the overlap. The HCF is 4a².
The HCF of 36q³r² and 48qr² — is it 12qr?
No. 12qr is a common factor, but not the highest. Both terms contain r², so the HCF is 12qr².
What does it mean to factorise an expression?
Write it as a product: the HCF outside a bracket, times what's left of each term inside. 6a − 3b + 12 = 3(2a − b + 4).
How do you check a factorisation?
Expand the bracket. You should get back exactly the expression you started with.
How do you know an expression is fully factorised?
The terms left inside the bracket share no common factor (other than 1).
Is 2(3y + 9) a correct way to factorise 6y + 18?
It's correct, but not fully factorised: 3y and 9 still share 3. Fully factorised, 6y + 18 = 6(y + 3).
Factorise fully: 12y² + 18y − 15xy
3y(4y + 6 − 5x). 3y is in all three terms; 4y and 6 share 2, but 5x doesn't, so nothing else can come out.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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