Unit 3 · Topic 06 · Algebra
Arjun divided 12x³ by 4x and got 3x² in two seconds. Then Meera handed him (x² + 5x + 6) ÷ (x + 2), and he froze.
"You already know how to do this," she said. "You just factorised it yesterday."
Arjun looked at the numerator, and the factorisation clicked into place.
Division of monomials was the easy part, and Arjun already had it down: 12x³ ÷ 4x meant dividing the numbers (12÷4=3) and subtracting the exponents of matching letters (x³÷x=x²), giving 3x². He'd checked it by multiplying back: 3x² × 4x = 12x³.
Then Meera wrote (x² + 5x + 6) ÷ (x + 2) and watched him stall. "You don't divide a polynomial by another polynomial term by term the way you did with monomials," she said. "But you already have a tool for exactly this shape."
Arjun looked at the numerator again. x² + 5x + 6 — two numbers that multiply to 6 and add to 5. That was 2 and 3. "It factors to (x+2)(x+3)," he said.
"Now divide," Meera said. Arjun wrote (x+2)(x+3) over (x+2), and the (x+2) in the top cancelled with the (x+2) on the bottom, the same way 6/2 cancels a shared factor of 2, leaving just (x+3).
"So dividing one polynomial by another is really: factorise the top, then cancel whatever matches the bottom," Arjun said. He checked by multiplying his answer back by the divisor: (x+3)(x+2) = x²+5x+6. It matched.
Meera gave him one more: (2x² + 7x + 3) ÷ (2x + 1). He factorised the top using the splitting-the-middle-term method from last session — first×last=6, pair 1 and 6 — getting (2x+1)(x+3). The (2x+1) cancelled with the divisor, leaving (x+3).
"What if the bottom doesn't cancel cleanly?" Arjun asked. "Then there's a remainder," Meera said, "same as dividing 7 by 2 leaves a remainder of 1 — but for now, every problem you'll meet factorises evenly, so cancelling is the whole method."
By the end, Arjun had a clear two-step habit: factorise the numerator completely first, then cancel any factor that matches the denominator exactly.
To divide one monomial by another, divide the coefficients and subtract the exponents of any matching letters — the reverse of multiplying monomials.
To divide a polynomial by a monomial, divide EVERY term of the polynomial by that monomial separately, the same distributing idea used when multiplying.
To divide a polynomial by a binomial (or another polynomial), first factorise the polynomial completely, then cancel any factor that exactly matches the divisor — this is why factorisation is worth mastering before division.
Always check a division by multiplying the quotient back by the divisor; the result must reproduce the original expression (for polynomial division with a remainder, adding the remainder back in as well).
Divide 15x⁴ by 5x² and check by multiplying back. Then divide (6x² + 9x) by 3x and check the same way.
Factorise x² − 4 and x² + 6x + 8, then divide each by (x + 2), noticing the same factor cancels in both.
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