Binomial Expansion

Maths: Edexcel A Level Pure Maths, topic 4.1.3

Binomial Expansion

We can work out the expansion of binomials using an equation involving factorials.

$Expanding binomials$

Expanding binomials

We have seen how we can expand binomials using the numbers in each row of Pascal's triangle.
We've also seen how to work out the numbers in Pascal's triangle using factorials.
This leads us to an equation we can use to expand binomials without having to think about Pascal's triangle.
This is called the binomial expansion or binomial theorem.

Binomial expansion equation

Binomial expansion equation

The binomial expansion of $(a+b)^n$ is:
$(a+b)^n=a^n + {n\choose 1}a^{n-1}b + {n\choose 2}a^{n-2}b^2 + ...+{n \choose r}a^{n-r}b^{r}+...+ b^n$
This equation applies when $n \in$ ℕ (is a positive whole number).
This equation is included in the formula booklet in the exam.

General term

General term

The general term in a binomial expansion is given by ${n \choose r}a^{n-r}b^{r}$ .
We can use this to find coefficients of specific orders of variables in the binomial expansion.

Example

Example

Use the binomial theorem to find the expansion of $(1-6x)^5$ .

Comparing variables

Comparing variables

If we compare the expression we want to expand with the equation given in the formula booklet:
$a = 1, b = -6x$ and $n = 5$ .
So we can substitute these into each term to find the binomial expansion.

First term

First term

The first term of the binomial expansion is $a^n$ .
Substituting in $a = 1$ and $n = 5$ , we have:
$a^n = 1^5 = 1$

Second term

Second term

The second term of the binomial expansion is ${n\choose 1}a^{n-1}b$ .
Substituting in for $a,b$ and $n$ , we have:
${n\choose 1}a^{n-1}b = {5\choose 1}1^4(-6x)$
Simplifying this term gives:
$\frac{5!}{1!\times4!}\times(-6x) = -30x$

Third term

Third term

The third term of the binomial expansion is ${n\choose 2}a^{n-2}b^2$ .
Substituting in for $a,b$ and $n$ , we have:
${n\choose 2}a^{n-2}b^2={5\choose 2}1^{3}(-6x)^2$
Simplifying this term gives:
$\frac{5!}{2!\times3!}\times36x^2 = 10\times36x^2 = 360x^2$

Fourth term

Fourth term

The fourth term of the binomial expansion is ${n\choose 3}a^{n-3}b^3$ .
Substituting in for $a,b$ and $n$ , we have:
${n\choose 3}a^{n-3}b^3={5\choose 3}1^{2}(-6x)^3$
Simplifying this term gives:
$\frac{5!}{3!\times2!}\times(-216)x^3 = 10\times(-216)x^3 = -2160x^3$

Fifth term

Fifth term

The fifth term of the binomial expansion is ${n\choose 4}a^{n-4}b^4$ .
Substituting in for $a,b$ and $n$ , we have:
${n\choose 4}a^{n-4}b^4 = {5\choose 4}ab^4$
Simplifying this term gives:
$\frac{5!}{4!\times1!}\times1296x^4 = 5\times1296x^4 = 6480x^4$

Final term

Final term

The sixth term of the binomial expansion is ${n\choose 5}a^{n-5}b^5$ .
We can tell that this is the final term in the expansion because there will be no powers of $a$ left in the term and that ${5\choose 5}=1$ .
So this term is simply equal to $b^n$ .
Substituting in for $b$ and $n$ , we have:
$b^n = (-6x)^5 = -7776x^5$

Sum the terms

Sum the terms

The expansion of $(1-6x)^5$ is equal to the sum of the terms in the binomial expansion, so the answer is:
$(1-6x)^5 = 1 -30x + 360x^2 -2160x^3 + 6480x^4-7776x^5$

Test yourself on Binomial Expansion

For which values of $n$ does the following binomial expansion apply?

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