Binomial Theorem • Topic 1 of 4

Expansion

The expansion of (a+b)ⁿ for a positive integer n is the sum nC0·aⁿ + nC1·a^(n−1)·b + nC2·a^(n−2)·b² + … + nCn·bⁿ. Three things stay constant and are worth burning in: there are exactly (n+1) terms; the power of a falls from n to 0 while the power of b climbs from 0 to n, so the two exponents in every term add up to n; and the coefficients are the binomial coefficients, which are symmetric (nCr = nC(n−r)) — the row reads the same forwards and backwards. For small n you can lift the coefficients straight from Pascal’s triangle, but the nCr formula scales to any n. A frequent CAT subtlety: (a−b)ⁿ has the same magnitudes but alternating signs (+,−,+,−…), so the odd-position terms flip. Always read off n correctly before you start.

✅ Solved examples

1. Write the expansion of (x + 2)⁴.
Coefficients 1,4,6,4,1. (x+2)⁴ = x⁴ + 4x³·2 + 6x²·4 + 4x·8 + 16 = x⁴ + 8x³ + 24x² + 32x + 16.
2. How many terms are there in the expansion of (3a − 2b)¹⁵?
A binomial to the power n has (n+1) terms. Here n = 15 ⇒ 16 terms.
3. In (a − b)⁶, what is the sign of the term containing b³?
b³ comes from r = 3, with factor (−1)³ = −1. The term is negative.
4. Find the coefficient of x² in (1 + x)⁵.
Coefficient of x² is 5C2 = 10.

✏️ Practice — try these, take hints as needed

1. How many terms are in the expansion of (2x + y)²⁰?
Terms = n + 1.
n = 20.
Add one.
21
2. Expand (x − 1)³.
Coefficients 1,3,3,1.
Signs alternate for (x−1).
x³ − 3x² + 3x − 1.
x³ − 3x² + 3x − 1
3. In (1 + x)⁷, find the coefficient of x³.
Coefficient is 7Cr with r = 3.
7C3 = 35.
No other factor since base is 1.
35
4. The exponents of a and b in any term of (a+b)ⁿ sum to what?
Power of a falls, power of b rises.
(n−r) + r.
They always total n.
n
5. What is the last term of (2x + 3)⁵?
Last term is r = n = 5.
nCn = 1, a^0 = 1.
3⁵.
243

📝 Topic test — 8 questions

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