Expanding takes a bracket and spreads it out. Factorising walks it backwards — pulling out what every piece shares and folding the bracket back on.
Pull it apartYou already know how to expand: 3(2x + 3) becomes 6x + 9. Factorising is the same road driven the other way — you start with 6x + 9 and figure out that the tidy, repackaged version is 3(2x + 3).
The trick is spotting the number that every part of the expression is secretly built from — the common factor — and lifting it out to the front of a bracket. Same value, neater outfit.
Before you touch the tool, here are the three words we'll use, defined in plain language the first time you meet them:
Term — one of the separate pieces of an expression, the bits joined by + or −. In 6x + 9 the terms are 6x and 9. Two terms, two pieces.
Common factor — a number that divides evenly into every term. Both 6 and 9 can be split into groups of 3 (6 = 3×2, 9 = 3×3), so 3 is a common factor of 6x + 9. So is 1, but 1 never helps — it pulls out nothing.
Factorise — to rewrite an expression as a factor multiplied by a bracket, like turning 6x + 9 into 3(2x + 3). You factorise fully when you've pulled out the highest common factor, so nothing more can come out of the bracket.
Pick an expression, then use the stepper to choose a number to pull out. Watch the tiles: when your number divides both terms, they snap into equal rows. Reach the highest common factor and the whole thing folds into a bracket.
Pulling out the highest common factor is called factorising fully — the numbers left inside share nothing but 1.
Every factorisation follows one recipe: find the biggest number that goes into every term, put it out front, and write what's left inside the bracket. Here it is on our three expressions — try each one in the tool above as you read.
6 and 9 both split into 3s. Pull out 3: 6x ÷ 3 = 2x, and 9 ÷ 3 = 3. The bracket is (2x + 3). Check by expanding: 3 × 2x = 6x, 3 × 3 = 9. Back where we started.
2 goes into both, but 4 is bigger and still goes into both. Pull out the biggest — 4. Then 4x ÷ 4 = x (just one x), and 8 ÷ 4 = 2. So 4(x + 2). If you'd stopped at 2(2x + 4), you could still pull a 2 out of the bracket — not fully factorised yet.
10 and 15 don't share a 2 (15 is odd) or a 10, but they both share 5. Pull out 5: 10x ÷ 5 = 2x, 15 ÷ 5 = 3. That gives 5(2x + 3), and 2 and 3 share nothing but 1 — so you're done.
A common factor snaps the tiles into equal rows. But if you pulled out a small common factor, the numbers still stuck inside the bracket will share something — meaning the bracket can be squeezed again. Factorising fully means going straight to the highest common factor (sometimes written HCF), so nothing is left to pull.
This second tool lights up the factors of each term. Where the two ladders share a rung, that number divides both — and the tallest shared rung is your highest common factor.
The tallest rung shared by both ladders is the highest common factor — pull that one out.
Pick the version that's factorised all the way — the highest common factor out front, nothing left to pull from the bracket.
Find the highest number that divides every term.
Write it in front of a bracket.
Put what's left of each term inside.