SL 1.9—Binomial theorem where n is an integer: IB DP Math AA Notes

What is the Binomial Theorem?

The binomial theorem provides a systematic method for expanding expressions of the form $(a + b)^{n}$ , where $n$ is a positive integer. Rather than multiplying out brackets repeatedly (imagine expanding $(a + b)^{7}$ by hand!), the theorem gives us a direct formula.

Binomial Theorem

For any positive integer $n$ , the expansion of $(a + b)^{n}$ is given by:

(a + b)^{n} = k = 0 \sum n (k n) a^{n - k} b^{k}

This produces $n + 1$ terms in total.

Written out fully, this looks like:
$(a + b)^{n} = (0 n) a^{n} + (1 n) a^{n - 1} b + (2 n) a^{n - 2} b^{2} + \dots + (n - 1 n) a b^{n - 1} + (n n) b^{n}$

Notice the pattern:

The powers of $a$ decrease from $n$ down to $0$
The powers of $b$ increase from $0$ up to $n$
The powers of $a$ and $b$ in each term always sum to $n$
The coefficients $(k n)$ are called binomial coefficients

Note

The binomial theorem formula is given in the IB formula booklet. However, you still need to understand how to apply it correctly — the formula alone won't expand the brackets for you.

Free preview

What is the Binomial Theorem?

9 more sections in this topic