Use the binomial theorem to expand (a plus b) to the power n for a positive integer n, using binomial coefficients

What this dot point is asking

SEAB wants you to expand a power of a bracket, $(a + b)^n$ for a positive whole number $n$, using the binomial theorem rather than multiplying the bracket out many times. You need the binomial coefficients (from Pascal's triangle or the $\binom{n}{r}$ notation) and the pattern of decreasing and increasing powers. This is the foundation for finding a single specific term, covered in the next dot point.

The answer

The pattern of an expansion

When you expand $(a + b)^n$, every term is a coefficient times a power of $a$ times a power of $b$. As you move along the expansion:

• the power of $a$ decreases from $n$ down to $0$,
• the power of $b$ increases from $0$ up to $n$,
• the two powers in each term always add up to $n$.

There are $n + 1$ terms in total.

Binomial coefficients from Pascal's triangle

The coefficients are the numbers in Pascal's triangle, where each number is the sum of the two above it:

• Row $n = 2$: $1,\ 2,\ 1$
• Row $n = 3$: $1,\ 3,\ 3,\ 1$
• Row $n = 4$: $1,\ 4,\ 6,\ 4,\ 1$
• Row $n = 5$: $1,\ 5,\ 10,\ 10,\ 5,\ 1$

For small $n$, reading off the right row is the fastest method.

The general formula

For larger $n$, the coefficients are written $\binom{n}{r}$ (read "$n$ choose $r$"), and the theorem is:

where $\binom{n}{r}$ can be found on your calculator (often labelled $nCr$).

Expanding a numerical bracket

When $a$ or $b$ is a number, raise it to the right power in each term and simplify. Keep track of powers carefully, for example $(2 + x)^3$ uses $2^3, 2^2, 2^1, 2^0$ as the power of $a$ falls.

Examples in context

Example 1. Approximating a value. Expanding $(1 + x)^n$ for small $x$ and keeping only the first few terms gives a quick approximation, for example $(1.02)^5 \approx 1 + 5(0.02) = 1.10$. The early terms of a binomial expansion are the basis of such estimates.

Example 2. Probability with repeated trials. The coefficients $\binom{n}{r}$ count the ways an outcome can happen across $n$ trials, which is why the same binomial numbers appear in the binomial probability distribution. Expanding the bracket and counting arrangements are two sides of the same idea.

Try this

Q1. Write down the coefficients for the expansion of a bracket to the power $5$. [1 mark]

• Cue. From Pascal's triangle: $1, 5, 10, 10, 5, 1$.

Q2. Expand $(1 + x)^3$. [2 marks]

• Cue. $1 + 3x + 3x^2 + x^3$.

Q3. Expand $(1 - 2x)^2$. [2 marks]

• Cue. $1 - 4x + 4x^2$, since $(-2x)^2 = 4x^2$.