GCSE-readyBinomial expansions: (a + bx)n and products
Most binomial questions are not simply (1 + x)n. You need to handle a number in front of the 1, a product of two brackets, and unknown values of n.
(a + bx)n with n a positive integer
(2 + x)5: the x2 term is 5C2 × 23 × x2 = 10 × 8x2 = 80x2.
(3 − 2x)4: the x3 term is 4C3 × 3 × (−2x)3 = -96x3. Keep the minus sign inside the bracket.
Rational n: take a factor out first
(4 + x)12 = 412(1 + x4)12 = 2(1 + x8 − ...) = 2 + x4 − ...
It is valid when |x4| < 1, that is |x| < 4.
Products and unknowns
x terms: 1 × (−6x) + 2x × 1 = -4x. x2 terms: 15 + 2 × (−6) = 3.
If (1 + kx)n has x coefficient 12 and x2 coefficient 60: nk = 12 and n(n − 1)2k2 = 60. Dividing gives (n − 1)k = 10, so k = 2 and n = 6.
Find the coefficient of x2 in (1 + 2x)(1 − x)6.
- (1 − x)6 = 1 − 6x + 15x2 − ...
- x2 terms: 1 × 15x2 + 2x × (−6x)
- = 15x2 − 12x2 = 3x2
Answer: 3
Include the powers of both terms in (a + bx)n. For rational n, factor out an first; the expansion is valid for |bx/a| < 1. For a product, collect every way to make the power you want.
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Find the coefficient of x2 in the expansion of (3 + x)4.
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Answer: 54
4C2 × 32 × (1)2 = 6 × 9 × 1 = 54.
Find the coefficient of x3 in the expansion of (2 − x)5.
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Answer: -40
5C3 × 22 × (−1)3 = 10 × 4 × (−1) = −40.
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Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.