How do I treat two bases?

Raise both bases to the same powers, so (2 + 3x)⁵ needs 2 cubed and 3 squared.

How do I find a constant term?

Set the power of x in the general term to zero, solve for r and substitute it back.

S6 Maths: Binomial Theorem, Paper 4 — coefficients, constant terms and products

2026-11-02

Paper 4 drills coefficient arithmetic from expansions to constant terms and products of two brackets, always via the general term.

Focus　Write the general term and match the power, raise both bases, and pair terms when two brackets multiply.

01　Paper details (check answers afterwards)

Item	Detail
Questions	10
Suggested time	25
Difficulty mix	Foundation 4 · Intermediate 3 · Challenge 2 · Error redo 1
Chapter	Chapter: S6 Maths: Binomial Theorem
Level	S6

02　Questions (10 items)

FoundationExpand (3 + x)³.
FoundationFind the coefficient of x³ in (1 + x)⁶.
FoundationFind the coefficient of x² in (1 − 3x)⁴.
FoundationWrite down the third term of (x + 3)⁵.
IntermediateFind the coefficient of x² in (2 + 3x)⁵.
IntermediateFind the constant term in (x³ + 1/x)⁸.
IntermediateFind the coefficient of x³ in (1 + 2x)⁷.
ChallengeIf C(n, 2) = 66, find n.
ChallengeFind the coefficient of x⁴ in (1 + x)⁴(1 − x)⁴.
Error redoError: a student says the coefficient of x² in (1 + x)⁵ is 5. What is correct?

03　Answers and one-line notes

#	Answer	One-line note
1	27 + 27x + 9x² + x³	Pair powers of 3 with powers of x: 27, 27, 9 and finally 1.
2	20	The coefficient is 6 choose 3, which is 20, since powers of 1 change nothing.
3	54	Take 4 choose 2 times minus 3 squared, so 6 times 9 gives 54.
4	90x³	The third term takes r equal to 2, so 5 choose 2 times 9 gives 90 times x cubed.
5	720	It is 5 choose 2 times 2 cubed times 3 squared, so 10 times 8 times 9 gives 720.
6	28	The power is 24 minus 4r, which vanishes at r equal to 6, and 8 choose 6 is 28.
7	280	It is 7 choose 3 times 2 cubed, so 35 times 8 gives 280.
8	n = 12	Setting n(n minus 1) over 2 to 66 gives (n minus 12)(n plus 11), so n is 12.
9	6	The product is (1 minus x squared) to the fourth, so the x fourth power coefficient is 4 choose 2, which is 6.
10	10	The coefficient is 5 choose 2, which is 10; the 5 is only the power.

04　What to do after marking

After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
Repeat the same item two days later; move to the next paper only when both attempts are correct.
Parents only need the answer and note columns; there is no need to explain every item.

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