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The binomial expansion of (a + b)ⁿ · Fathom

The binomial expansion of (a + b)ⁿ

6 minAS Level1.6.1

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can expand $(a + b)^n$ for any positive integer $n$ , using Pascal's triangle or the binomial coefficients $\binom{n}{r}$ , and you know what $n!$ means.

Four brackets? No thanks

You could expand $(3 + 2x)^4$ by multiplying out four brackets. It works.

It also takes ages and one slip wrecks everything.

Look at the first few powers instead and a pattern jumps out:

$n$	$(a + b)^n$
1	$a + b$
2	$a^2 + 2ab + b^2$
3	$a^3 + 3a^2b + 3ab^2 + b^3$
4	$a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4$

Powers of $a$ go down by one each term; powers of $b$ go up by one.
In every term the powers add up to $n$ .
There are $n + 1$ terms.
The numbers in front, the coefficients, come from Pascal's triangle.

Quick check

Multiple choiceQuestion 1 · Recall

Answer the 2 questions above to continue.