How do I handle a minus sign?

Raise the minus sign together with the coefficient; an even power turns it positive.

How do I find a coefficient in a product?

List every pairing that gives the required power, multiply each and add the results.

S6 Maths: Binomial Theorem, Paper 3 — mixed terms and coefficients

2026-11-02

Tutor Prep Guide · Primary English

The general term pins down a chosen power, while products of two expansions need every pairing added up.

Focus　Write the general term and match the power; for two brackets list every pairing and add the products.

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 (1 + x)³.
FoundationFind the coefficient of x² in (1 + x)⁵.
FoundationFind the coefficient of x³ in (1 − 2x)⁴.
FoundationFind the coefficient of x² in (2 + x)⁴.
IntermediateFind the coefficient of x² in (1 + 3x)⁵.
IntermediateFind the constant term in (x − 2/x)⁴.
IntermediateThe coefficient of x² in (1 + x)ⁿ is 45. Find n.
ChallengeFind the coefficient of x² in (1 + x)⁵(1 − x)³.
ChallengeFind the constant term in (2x + 3/x)⁴.
Error redoError: a student gives 8 as the coefficient of x³ in (1 + 2x)⁵. What is correct?

03　Answers and one-line notes

#	Answer	One-line note
1	1 + 3x + 3x² + x³	The coefficients are 1, 3, 3 and 1 as the powers climb from zero to three.
2	10	The coefficient is simply five choose two, which is 10.
3	−32	Four choose three is 4, and minus 2 cubed is minus 8, giving minus 32.
4	24	Six ways times 2 squared gives 24.
5	90	Ten choose times 3 squared, namely 9, gives 90.
6	24	The power of x is 4 minus 2r, so r is 2 and the term is six times 4.
7	n = 10	Ten choose two is 45, so n is 10.
8	−2	Add the three ways of making x squared to get minus 2.
9	216	Setting the power 4 minus 2r to zero gives r equals 2, so the term is 6 times 4 times 9.
10	80	The binomial coefficient of 10 was left out, so the answer is 80.

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.

Book an S6 Maths tutor Tutor Registration →

Questions and answers are written originally by the TutorZone team and are not taken from any school, publisher or examination authority paper. Every value is checked by computer.