2. Binomial Expansion

Video coming soon2 worked examples

Theory

The Binomial Theorem lets us expand $(a+b)^n$ without multiplying out every bracket. For a positive integer $n$ :

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

Here $\binom{n}{r} = \dfrac{n!}{r!\,(n-r)!}$ is the binomial coefficient — the ${}^nC_r$ button on your calculator, or the entries of a row of Pascal's triangle. For example, the coefficients for $n=4$ are $1,\ 4,\ 6,\ 4,\ 1$ .

The Golden Rule: the power of the first term counts down from $n$ to $0$ , while the power of the second term counts up from $0$ to $n$ . In every term the two powers must add up to $n$ .

⚠️ Common Examiner Traps

Not raising the whole term to the power: $(2x)^3 = 8x^3$ , not $2x^3$ . The coefficient must be raised to the power as well as the variable.
Sign slips with a negative second term: in $(a-b)^n$ the signs alternate $+,\,-,\,+,\,-\dots$ — write the bracket as $(a+(-b))^n$ to keep track.
Miscounting the coefficients: use ${}^nC_r$ on the calculator rather than trying to extend Pascal's triangle from memory under exam pressure.

Worked examples

Example 1

Expand $(2x + 3)^4$ fully.

Step 1: Write out the expansion using $n=4$ , with the coefficients $1,4,6,4,1$ :

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

Step 2: Evaluate each power and coefficient:

= (16x^4) + 4(8x^3)(3) + 6(4x^2)(9) + 4(2x)(27) + 81

Step 3: Simplify:

= 16x^4 + 96x^3 + 216x^2 + 216x + 81

Example 2

Expand $\left(x - \dfrac{2}{x}\right)^4$ .

Step 1: Write the general term. With $a = x$ and $b = -\dfrac{2}{x}$ :

\binom{4}{r} x^{4-r}\left(-\frac{2}{x}\right)^r = \binom{4}{r}(-2)^r x^{4-2r}

Step 2: Evaluate for $r = 0,1,2,3,4$ :

\begin{aligned} r=0:&\ \ x^4 \\ r=1:&\ -8x^2 \\ r=2:&\ \ 24 \\ r=3:&\ -\frac{32}{x^2} \\ r=4:&\ \ \frac{16}{x^4} \end{aligned}

Step 3: Collect the terms:

\left(x - \frac{2}{x}\right)^4 = x^4 - 8x^2 + 24 - \frac{32}{x^2} + \frac{16}{x^4}