A-Level — mathsStandard2 min read

Binomial Expansion

Expand expressions using the binomial theorem at A-Level. Apply for positive integer powers and general binomial expansions.

Tutor AI Team2026-02-25T01:53:03.156Z

The binomial theorem provides a formula for expanding $(a + b)^n$ . At A-Level, you need both the standard expansion for positive integers and the general expansion for rational powers.

For Positive Integer $n$

$(a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r$

$\binom{n}{r} = \frac{n!}{r!(n-r)!}$

Example

$(1 + x)^4 = 1 + 4x + 6x^2 + 4x^3 + x^4$

$(2 + 3x)^3 = 8 + 36x + 54x^2 + 27x^3$

Finding a Specific Term

The $(r+1)$ th term: $\binom{n}{r} a^{n-r} b^r$ .

General Binomial Expansion ($n$ not a positive integer)

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

Valid for $|x| < 1$ .

Example

$(1 + x)^{-1} = 1 - x + x^2 - x^3 + ...$

$(1 + x)^{\frac{1}{2}} = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 - ...$

For $(a + bx)^n$

Factor out $a^n$ : $a^n(1 + \frac{bx}{a})^n$ , then expand, valid for $|\frac{bx}{a}| < 1$ .

Worked Example: Example 1

Problem

Expand $(1 - 2x)^{-3}$ up to $x^2$ .

$= 1 + (-3)(-2x) + \frac{(-3)(-4)}{2}(-2x)^2 + ...$ $= 1 + 6x + 24x^2 + ...$

Valid for $|2x| < 1$ , i.e., $|x| < \frac{1}{2}$ .

Solution

Practice Problems

1. Expand $(2 + x)^5$ .
1. Find the coefficient of $x^3$ in $(1 - 3x)^8$ .
1. Expand $(1 + 4x)^{\frac{1}{2}}$ up to $x^3$ and state validity.

Want to check your answers and get step-by-step solutions?

Key Takeaways

✓
Integer $n$ : use $\binom{n}{r}$ , finite expansion.
✓
General $n$ : infinite series, valid for $|x| < 1$ .
✓
For $(a+bx)^n$ : factor out $a^n$ first.
✓
State the range of validity.

Tags:

#a-level#maths#pure#algebra#binomial

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For Positive Integer $n$

Example

Finding a Specific Term

General Binomial Expansion ($n$ not a positive integer)

Example

For (a+bx)n(a + bx)^n(a+bx)n

Worked Example: Example 1

Practice Problems

Key Takeaways

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For $(a + bx)^n$