What is the binomial theorem and how is it used in A-Level Maths?

The binomial theorem states that (a+b)^n = sum of C(n,r) * a^(n-r) * b^r for r from 0 to n, where C(n,r) = n! / (r!(n-r)!) is the binomial coefficient. At A-Level, it appears in Pure Maths (Year 1 and Year 2). You use it to expand expressions like (1+2x)^5 or (3-x)^4 without multiplying brackets repeatedly. It is assessed by AQA, Edexcel and OCR exam boards.

How do I read Pascal's triangle for binomial expansions?

Pascal's triangle gives the binomial coefficients for each power n. Row 0 is just 1. Row 1 is 1,1. Row 2 is 1,2,1. Row 3 is 1,3,3,1. Row 4 is 1,4,6,4,1. Each number is the sum of the two numbers directly above it. For (a+b)^n, read off row n for the coefficients. For example, (a+b)^4 = 1a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + 1b^4.

What is the general term in a binomial expansion?

The general term (also called the (r+1)th term) in the expansion of (a+b)^n is: T(r+1) = C(n,r) * a^(n-r) * b^r. To find a specific term, substitute the value of r. For example, in (2+x)^6, the 3rd term (r=2) is C(6,2) * 2^4 * x^2 = 15 * 16 * x^2 = 240x^2. This is a common A-Level exam question type.

When can I use the binomial approximation (1+x)^n for small x?

The binomial series expansion (1+x)^n = 1 + nx + n(n-1)x^2/2! + n(n-1)(n-2)x^3/3! + ... is valid for |x| < 1 when n is not a positive integer (i.e., for fractional or negative powers). For positive integer n, the expansion is exact and terminates. When |x| is small (much less than 1), you can approximate (1+x)^n ≈ 1 + nx for a quick estimate. This approximation is used in physics and engineering for small perturbations.

How do I find the coefficient of a specific term like x^3 in an expansion?

To find the coefficient of x^k in the expansion of (a + bx)^n, set r = k in the general term T(r+1) = C(n,r) * a^(n-r) * (bx)^r. The coefficient is C(n,k) * a^(n-k) * b^k. For example, to find the coefficient of x^3 in (2+3x)^7: T(4) = C(7,3) * 2^4 * (3x)^3 = 35 * 16 * 27x^3 = 15120x^3. Always check which term contains x^k by matching powers.

What is the difference between C(n,r), nCr and the choose notation in A-Level?

All three notations mean the same thing: the number of ways to choose r items from n items without regard to order. C(n,r) = nCr = (n choose r) = n! / (r!(n-r)!). At A-Level, your calculator will have an nCr button. For example, C(5,2) = 5!/(2!3!) = 10. These are the binomial coefficients that appear as the middle row entries in Pascal's triangle. In statistics, nCr is used for combinations and probability calculations.

How do negative or fractional powers work in binomial expansion at A-Level Year 2?

At A-Level Year 2 (A2), the binomial series is extended to negative and fractional values of n. The expansion (1+x)^n = 1 + nx + n(n-1)x^2/2! + n(n-1)(n-2)x^3/3! + ... holds for |x| < 1 with non-integer n. For example, (1+x)^(1/2) = 1 + x/2 - x^2/8 + x^3/16 - ... This infinite series converges for |x| < 1. You must always state the range of validity. First rewrite expressions like (2+x)^(-1) as 2^(-1)(1+x/2)^(-1) before expanding.

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Binomial Expansion: (a + b)^n

Enter the terms and power to get the full expansion with step-by-step working.

First term (a) — e.g. 2x, 3, x

Second term (b) — e.g. 2, -y, 3x

Power (n) — positive integer 1–10

What is the Binomial Theorem?

The binomial theorem is a fundamental result in algebra that gives a formula for expanding expressions of the form (a + b)^n, where n is a positive integer. It states:

(a + b)^n = ∑[r=0 to n] C(n,r) × a^(n-r) × b^r

where C(n,r) = n! / (r! × (n-r)!) is the binomial coefficient, read as "n choose r". This expansion produces n+1 terms. The binomial theorem is a core topic in A-Level Pure Mathematics and appears in both AS and A2 papers across AQA, Edexcel and OCR specifications.

Pascal's Triangle and Binomial Coefficients

Pascal's triangle provides a visual and recursive way to find binomial coefficients without computing factorials. Each row n contains the coefficients for the expansion of (a+b)^n:

Row 0: 1 → (a+b)^0 = 1
Row 1: 1, 1 → (a+b)^1 = a + b
Row 2: 1, 2, 1 → (a+b)^2 = a² + 2ab + b²
Row 3: 1, 3, 3, 1 → (a+b)^3 = a³ + 3a²b + 3ab² + b³
Row 4: 1, 4, 6, 4, 1 → (a+b)^4 = a&sup4; + 4a³b + 6a²b² + 4ab³ + b&sup4;

Each entry is formed by adding the two entries directly above it. The triangle has many remarkable properties: the rows sum to powers of 2, and the diagonals contain triangular numbers, Fibonacci numbers (in a diagonal sum), and more.

How to Expand (a + b)^n Step by Step

Follow these steps to expand any binomial expression at A-Level:

Identify a, b and n from the expression. For (2x + 3)^5, a = 2x, b = 3, n = 5.
Write out each term using T(r+1) = C(n,r) × a^(n-r) × b^r for r = 0, 1, 2, …, n.
Read off the coefficients from row n of Pascal's triangle or compute nCr directly.
Simplify each term by collecting numerical coefficients and simplifying powers.
Write the final answer in ascending or descending powers of the variable.

Worked Example: Expand (3 + x)^4
Using row 4 of Pascal's triangle: 1, 4, 6, 4, 1
T1: C(4,0) × 3^4 × x^0 = 1 × 81 = 81
T2: C(4,1) × 3^3 × x^1 = 4 × 27 × x = 108x
T3: C(4,2) × 3^2 × x^2 = 6 × 9 × x^2 = 54x^2
T4: C(4,3) × 3^1 × x^3 = 4 × 3 × x^3 = 12x^3
T5: C(4,4) × 3^0 × x^4 = 1 × x^4 = x^4
Answer: (3 + x)^4 = 81 + 108x + 54x² + 12x³ + x&sup4;

The General Term Formula

The general term (the (r+1)th term) in the expansion of (a+b)^n is:

T(r+1) = C(n,r) × a^(n-r) × b^r

This formula is essential for A-Level exam questions that ask you to "find the term in x^k" or "find the coefficient of x^3". You set the power of x equal to k and solve for r.

Binomial Approximation for Small x

When n is a non-integer (fractional or negative) and |x| < 1, the binomial series gives an infinite expansion:

(1+x)^n = 1 + nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3! + …

This series converges for |x| < 1. For very small x, only the first two or three terms are needed for a good approximation. Common examples at A-Level Year 2 include (1+x)^(1/2) (square root approximation) and (1+x)^(-1) (related to geometric series). Always state the range of validity: |x| < 1.

Key Formulae Summary for A-Level

Formula	Expression
Binomial Theorem	(a+b)^n = ∑ C(n,r) a^(n-r) b^r
Binomial Coefficient	C(n,r) = n! / (r!(n-r)!)
General Term	T(r+1) = C(n,r) a^(n-r) b^r
Sum of Coefficients	2^n (set a=b=1)
Binomial Series (\|x\|<1)	(1+x)^n = 1 + nx + n(n-1)x^2/2! + …

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Binomial Expansion: (a + b)^n

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(1+x)^n Approximation (small x)

Pascal's Triangle