For students and parents · O-Level A-Math
The Binomial Theorem, From the General Term to the Trap
Expanding a power of a bracket using the general term.
Mrs Eileen Toh
Founder & Curriculum Architect · ex-MOE · 3 min read · Updated 29 Jun 2026
The binomial theorem lets you expand something like (2 + x)5 without multiplying five brackets by hand, and more usefully, lets you reach straight for one term you want without writing the whole expansion. It appears in every A-Math paper GPA has counted, and most of its marks are lost not to the idea but to one careless step in the index arithmetic.
The general term is the whole tool
For (a + b)n, the term containing br is
C(n, r) x an-r x br,
where C(n, r) is the binomial coefficient, the number from Pascal's triangle or the nCr button. The power of a and the power of b always add to n. That single line is the entire theorem; everything else is choosing r.
Expanding in full
(2 + x)5 = 32 + 80x + 80x2 + 40x3 + 10x4 + x5. Each coefficient is C(5, r) x 25-r: the x3 term, for instance, is C(5, 3) x 22 = 10 x 4 = 40. You rarely need the full expansion, which is the point of the next two skills.
Picking out one coefficient
Find the coefficient of x2 in (1 + 2x)6.
You want the term with br = (2x)2, so r = 2:
C(6, 2) x 14 x (2x)2 = 15 x 1 x 4x2 = 60x2.
The coefficient is 60. Note where the trap sits: the 2 inside (2x)2 must be squared too. Students who write C(6, 2) = 15 and stop have dropped the 22, and the mark with it.
Finding the independent term
Find the term independent of x in (x + 2/x2)6.
The general term is C(6, r) x x6-r x (2/x2)r = C(6, r) x 2r x x6 - 3r. The term is independent of x when the power of x is zero: 6 - 3r = 0, so r = 2. That term is C(6, 2) x 22 = 15 x 4 = 60. The skill is to write the power of x as a single expression in r, set it to what you need, and solve for r.
The teaching point
The binomial theorem is one formula, the general term, used three ways: expand everything, reach for one coefficient, or solve for the term you want. The marks leak at the index arithmetic, forgetting to raise the inner coefficient to the power, or mis-collecting the power of x. The fix is to write the general term out in full before substituting, so the powers are visible and the inner coefficient cannot be missed.
If your child can quote the formula but slips when a coefficient or a fraction sits inside the bracket, that is a precise, common gap. GPA's Secondary Math programme drills the general term on real A-Math papers until the index step is automatic. A short diagnostic consult will show you where it breaks.
Every coefficient on this page was checked independently before publishing.