Cambridge International AS and A Level Mathematics Pure Mathematics 1

Binomial expansions

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Binomial expansions

A special type of series is produced when a binomial (i.e. two-part) expression

like (x + 1) is raised to a power. The resulting expression is often called a

binomial expansion.

The simplest binomial expansion is (x + 1) itself. This and other powers of

(x + 1) are given below.

(x + 1)^1 = 1 x + 1

(x + 1)^2 = 1 x^2 + 2 x + 1

(x + 1)^3 = 1 x^3 + 3 x^2 + 3 x + 1

(x + 1)^4 = 1 x^4 + 4 x^3 + 6 x^2 + 4 x + 1

(x + 1)^5 = 1 x^5 + 5 x^4 + 10 x^3 + 10 x^2 + 5 x + 1

If you look at the coefficients on the right-hand side above you will see that they

form a pattern.

(1)

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

This is called Pascal’s triangle, or the Chinese triangle. Each number is obtained by

adding the two above it, for example

4 + 6

gives 10

This pattern of coefficients is very useful. It enables you to write down the

expansions of other binomial expressions. For example,

(x + y) = 1 x + 1 y

(x + y)^2 = 1 x^2 + 2 xy + 1 y^2

(x + y)^3 = 1 x^3 + 3 x^2 y + 3 xy^2 + 1 y^3

ExamPlE 3.11 Write out the binomial expansion of (x + 2)^4.

SOlUTION

The binomial coefficients for power 4 are 1 4 6 4 1.

In each term, the sum of the powers of x and 2 must equal 4.

So the expansion is

1 × x^4 + 4 × x^3 × 2 + 6 × x^2 × 22 + 4 × x × 23 + 1 × 24

i.e. x^4 + 8 x^3 + 24 x^2 + 32 x + 16.

Expressions like these,

consisting of integer

powers of x and constants

are called polynomials.

These numbers are called

binomial coefficients.

Notice how in each term

the sum of the powers of

x and y is the same as the

power of (x + y).

These numbers are called

This is a binomial expression. binomial coefficients.