8.3PureStretch

The Binomial Expansion

The binomial theorem expands $(a + b)^{n}$ for any power using the choose function — no triangle required. Master the general formula and you can expand huge powers, or pick out a single term, in seconds.

35 min Video by Zeeshan Zamurred The Binomial Expansion

Edexcel AS Level Maths: 8.3 The Binomial ExpansionWatch the full walkthrough before the notes below.

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What you'll be able to do

State the binomial theorem
Expand (a + b)ⁿ using binomial coefficients
Expand brackets such as (1 + x)ⁿ and (2 − 3x)ⁿ
Handle coefficients and signs accurately

The binomial theorem

Each term uses a binomial coefficient, a decreasing power of the first term and an increasing power of the second. The powers in each term always add up to $n$ .

(a + b)^{n} = r = 0 \sum n (r n) a^{n - r} b^{r}

Coefficient

(r n)

, power of

a

falling, power of

b

rising.

Expanding (1 + x)ⁿ

The most common form simplifies nicely because powers of $1$ disappear, leaving a tidy series in $x$ .

(1 + x)^{n} = 1 + n x + \frac{n ( n - 1 )}{2 !} x^{2} + \frac{n ( n - 1 ) ( n - 2 )}{3 !} x^{3} + \dots

A widely used special case worth memorising.

1Coefficients (row 4):

1, 4, 6, 4, 1

Answer

1 + 4 x + 6 x^{2} + 4 x^{3} + x^{4}

Brackets with coefficients and signs

When a term has a coefficient or a minus sign, substitute the $whole$ term (including its power) into each part of the expansion. Brackets around the substituted term prevent errors.

1Coefficients (row 4):

1, 4, 6, \dots

2Terms:

2^{4} + 4 (2^{3}) (- 3 x) + 6 (2^{2}) (- 3 x)^{2}

3Evaluate:

16 + 4 (8) (- 3 x) + 6 (4) (9 x^{2})

Answer

16 - 96 x + 216 x^{2}

Tip — Always bracket the substituted term: (−3x)² = 9x², not −9x².

Formula recap

(a + b)^{n} = r = 0 \sum n (r n) a^{n - r} b^{r}

The binomial theorem.

(1 + x)^{n} = 1 + n x + \frac{n ( n - 1 )}{2 !} x^{2} + \dots

Special case (1 + x)ⁿ.

powers in each term sum to n

Quick self-check.

Common mistakes to avoid

Squaring −3x as −9x².

Bracket the whole term: (−3x)² = 9x².

Forgetting to raise the coefficient (e.g. the 2 in (2 − 3x)⁴) to its power.

Each term needs the full power of the first term, e.g. 2³, 2².

Key takeaways

(a + b)ⁿ = Σ nCr · aⁿ⁻ʳ · bʳ; powers of a fall, powers of b rise.
(1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + … is the key special case.
Bracket substituted terms so coefficients and signs are handled correctly.
Check: the two powers in every term add to n.

Test yourself

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