Unit 3 · Topic 03 · Algebra
Arjun tried to factorise 6x² + 9x and wrote 3(2x + 3x), then simplified it back to 6x² + 9x² and stared at the mismatch.
Meera pointed at his second term. "You multiplied the 3 back in wrong. Watch what happens if you pull it out of BOTH terms properly."
Arjun redid it, and this time multiplying back out gave him exactly what he'd started with.
Factorisation was the reverse of everything they'd just practised: instead of expanding brackets, they now had to find brackets that expand back to a given expression. Arjun's first attempt at 6x² + 9x went wrong because he'd pulled a 3 out of 6x² and left it as 2x, but pulled a 3 out of 9x and written 3x instead of 3 — an inconsistent split.
Meera made him check the greatest common factor properly first. 6x² has factors 2, 3, x, x. 9x has factors 3, 3, x. The factors common to both are 3 and one x, so the greatest common factor is 3x. Dividing 6x² by 3x gives 2x, and dividing 9x by 3x gives 3. So 6x² + 9x = 3x(2x + 3).
"Always check by expanding back," Meera said. Arjun multiplied 3x by 2x to get 6x², then 3x by 3 to get 9x, and added them: 6x² + 9x. It matched.
Then Meera gave him a harder one with four terms and no single common factor across all of them: xy + 3x + 2y + 6. Arjun tried pulling out an x from everything and got stuck — the 2y and 6 had no x in them.
"Group the terms in pairs instead," Meera said. She split it as (xy + 3x) + (2y + 6). The first pair has a common factor of x: x(y + 3). The second pair has a common factor of 2: 2(y + 3). Now both pairs share the exact same bracket, (y + 3), so that whole bracket can be pulled out: (y + 3)(x + 2).
Arjun checked by expanding: y·x + y·2 + 3·x + 3·2 = xy + 2y + 3x + 6, which reorders to the original. "So regrouping just turns a four-term problem into two two-term problems, each with its own common factor — as long as the leftover brackets match."
"That's the whole trick," Meera said. "If the brackets don't match after grouping, try pairing the terms differently before giving up."
By the end, Arjun had a rule he trusted: always look for one common factor across everything first, and only reach for grouping when that search comes up empty.
Factorisation means writing an expression as a product of factors, the reverse of expanding brackets. The first thing to try is always the greatest common factor (GCF): find the largest number and the highest power of each letter that divides every term, then pull it outside a bracket.
When there is no single common factor across all terms — often with four terms — try regrouping: split the expression into pairs, factor each pair separately, and check whether the two pairs now share an identical bracket. If they do, that bracket factors out too, leaving a two-factor answer.
A factorisation is only correct if expanding it back out reproduces the original expression exactly — this is the one check that never lies, so use it every time.
Regrouping sometimes needs the terms reordered first, since the natural order they're written in doesn't always group into matching brackets on the first try.
Factorise 4x + 8 and 10y − 15 by finding the GCF of each, then check both by expanding back.
Take the four terms ab + 4a + 3b + 12, try grouping them as written, and confirm both pairs leave the same bracket behind.
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