Binomial Expansion

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Binomial Expansion

Binomial expansion uses binomial coefficients to expand two terms in brackets of the form $(ax+b)^{n}$ . When $n$ is a positive whole number the expansion is finite. When $n$ is not, the expansion is infinite.

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Pascal’s Triangle

Pascal’s triangle is made by putting $1$ s down the sides, then every number comes from adding together the two above it.

The numbers on the $n$ th row are the coefficients of $(1+x)^{n}$

$(1+x)^{0}=1$

$(1+x)^{1}=1+x$

$(1+x)^{2}=1+2x+x^{2}$

$(1+x)^{3}=1+3x+3x^{2}+x^{3}$

$(1+x)^{4}=1+4x+6x^{2}+4x^{3}+x^{4}$

and so on.

Binomial Coefficients

The binomial coefficient $begin{pmatrix}nmend{pmatrix}=dfrac{n!}{m!(n-m)!}$ where $m$ and $n$ are whole numbers and $n>m$ . We can also write them as $text{}^{n}C_{m}$ .

The binomial coefficients are the same as the numbers in Pascal’s triangle.

This means we can write our expansion as:

$(1+x)^{n}=begin{pmatrix}n�end{pmatrix}+begin{pmatrix}n1end{pmatrix}x+begin{pmatrix}n2end{pmatrix}x^{2}+…+begin{pmatrix}nn-1end{pmatrix}x^{n-1}+begin{pmatrix}nnend{pmatrix}x^{n}$

General Binomial Expansion Formula

So far we have only seen how to expand $(1+x)^{n}$ , but ideally we want a way to expand more general things, of the form $(a+b)^{n}$ . In this expansion, the $m$ th term has powers $a^{m}b^{n-m}$ . We can use this, along with what we know about binomial coefficients, to give the general binomial expansion formula.

$(a+b)^{n}=begin{pmatrix}n�end{pmatrix}a^{n}+begin{pmatrix}n1end{pmatrix}a^{n-1}b+begin{pmatrix}n2end{pmatrix}a^{n-2}b^{2}+…+begin{pmatrix}nmend{pmatrix}a^{m}b^{n-m}+…+begin{pmatrix}nn-1end{pmatrix}ab^{n-1}+begin{pmatrix}nnend{pmatrix}b^{n}$

Example 1: Binomial Expansion

Expand $(1+2x)^{2}$

[2 marks]

$(1+2x)^{2}=begin{pmatrix}2�end{pmatrix}1^{2}+begin{pmatrix}21end{pmatrix}1times2x+begin{pmatrix}22end{pmatrix}(2x)^{2}$

Find the binomial coefficients:

$begin{pmatrix}2�end{pmatrix}=1$

$begin{pmatrix}21end{pmatrix}=2$

$begin{pmatrix}22end{pmatrix}=1$

Substitute binomial coefficients into equation:

$begin{aligned}(1+2x)^{2}&=1times1^{2}+2times1times2x+1times(2x)^{2}[1.2em]&=1times1+2times2x+1times4x^{2}[1.2em]&=1+4x+4x^{2}end{aligned}$

Example 2: Finding Coefficients

Find the coefficient of $x^{3}$ in $(x+8)^{6}$ .

[2 marks]

$x^{3}$ term is $begin{pmatrix}63end{pmatrix}x^{3}8^{6-3}$

$begin{pmatrix}63end{pmatrix}=20$

$8^{6-3}=8^{3}=512$

$x^{3}$ term is $512times20x^{3}$

$x^{3}$ term is $10240x^{3}$

Coefficient of $x^{3}$ is $10240$

Binomial Expansion Example Questions

Question 1: What is the eighth row of Pascal’s triangle?

(Hint: The triangle pictured on this page goes up to row seven.)

[2 marks]

$1$ s down the sides.

$1+6=7$ so $7$ is the next number in.

$6+15=21$ so $21$ is the next number in.

$15+20=35$ so $35$ is the next number in.

So the whole row is:

$1,7,21,35,35,21,7,1$

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Question 2: Expand $left(dfrac{1}{4}-2xright)^{4}$

[5 marks]

$left(dfrac{1}{4}-2xright)^{4}=begin{pmatrix}4end{pmatrix}(-2x)^{4}+begin{pmatrix}41end{pmatrix}dfrac{1}{4}(-2x)^{3}+begin{pmatrix}42end{pmatrix}left(dfrac{1}{4}right)^{2}(-2x)^{2}+begin{pmatrix}43end{pmatrix}left(dfrac{1}{4}right)^{3}(-2x)+begin{pmatrix}44end{pmatrix}left(dfrac{1}{4}right)^{4}$

$begin{pmatrix}4end{pmatrix}=1$

$begin{pmatrix}41end{pmatrix}=4$

$begin{pmatrix}42end{pmatrix}=6$

$begin{pmatrix}43end{pmatrix}=4$

$begin{pmatrix}41end{pmatrix}=1$

$left(dfrac{1}{4}-2xright)^{4}=big( 1times(-2x)^{4}big) +big( 4timesdfrac{1}{4}(-2x)^{3}big) +big( 6timesleft(dfrac{1}{4}right)^{2}(-2x)^{2}big) +big( 4timesleft(dfrac{1}{4}right)^{3}(-2x)big) +big( 1timesleft(dfrac{1}{4}right)^{4}big)$

$left(dfrac{1}{4}-2xright)^{4}=big( 1times16x^{4}big) +big( 4timesdfrac{1}{4}times(-8)x^{3}big) +big( 6timesdfrac{1}{16}times4x^{2}big) +big( 4timesdfrac{1}{64}times(-2)xbig) +big( 1timesdfrac{1}{256}big)$

$left(dfrac{1}{4}-2xright)^{4}=16x^{4}-8x^{3}+dfrac{24}{16}x^{2}-dfrac{8}{64}x+dfrac{1}{256}$

$left(dfrac{1}{4}-2xright)^{4}=16x^{4}-8x^{3}+dfrac{3}{2}x^{2}-dfrac{1}{8}x+dfrac{1}{256}$

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Question 3: What is the coefficient of $x^{4}$ in $(5x-2)^{9}$ ?

[3 marks]

$x^{4}$ term is $begin{pmatrix}94end{pmatrix}(5x)^{4}(-2)^{5}=-begin{pmatrix}94end{pmatrix}5^{4}x^{4}2^{5}$

$begin{pmatrix}94end{pmatrix}=126$

$5^{4}=625$

$2^{5}=32$

$x^{3}$ coefficient is $-126times625times32=-2520000$

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Question 4: Find $p$ such that the $x^{2}$ coefficient in $left(dfrac{2}{3}x+pright)^{3}$ is $4$ .

[3 marks]

$x^{2}$ coefficient is $begin{pmatrix}32end{pmatrix}left(dfrac{2}{3}right)^{2}p$

$begin{pmatrix}32end{pmatrix}=3$

$left(dfrac{2}{3}right)^{2}=dfrac{4}{9}$

Hence, $3timesdfrac{4}{9}times p=4$

$dfrac{4}{3}p=4$

$p=4timesdfrac{3}{4}$

$p=3$

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Additional Resources

Exam Tips Cheat Sheet

A Level

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Formula Booklet

A Level

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Specification Points Covered

D1 – Understand and use the binomial expansion of $(a+bx)^n$ for positive integer $n$ ; the notations $n!$ and $nCr$ ; link to binomial probabilities
Extend to any rational $n$ , including its use for approximation; be aware that the expansion is valid for $left|dfrac{bx}{a}right|<1$ . (proof not required)

Binomial Expansion Worksheet and Example Questions

Polynomials

Exam Questions Mark Scheme

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Try it yourself

Binomial Expansion

Binomial Expansion

A Level Maths Predicted Papers 2025

A-A* A Level Maths Practice Papers

Pascal’s Triangle

Binomial Coefficients

General Binomial Expansion Formula

Example 1: Binomial Expansion

Example 2: Finding Coefficients

Binomial Expansion Example Questions