Question

How do you use the binomial series to expand `(x+1+x^-1)^4`?

Answer 1

`(x+1+x^(-1))^4`

`=x^4+4x^3+10x^2+16x+19+16x^(-1)+10x^(-2)+4x^(-3)+x^(-4)`

Explanation:

This is a trinomial, not a binomial. With binomials you could use Pascal's triangle to give you the coefficients.

Since the terms of the given trinomial are in geometric progression, we can use coefficients from a generalisation of Pascal's triangle that adds together `3` terms rather than `2`:

`color(white)(00000000000000)1`

`color(white)(00000000000)1color(white)(00)1color(white)(00)1`

`color(white)(00000000)1color(white)(00)2color(white)(00)3color(white)(00)2color(white)(00)1`

`color(white)(00000)1color(white)(00)3color(white)(00)6color(white)(00)7color(white)(00)6color(white)(00)3color(white)(00)1`

`color(white)(00)1color(white)(00)4color(white)(0)10color(white)(0)16color(white)(0)19color(white)(0)16color(white)(0)10color(white)(00)4color(white)(00)1`

In this triangle each number is the sum of the three numbers above it: left, centre and right.

Hence we find:

`(x+1+x^(-1))^4`

`=x^4+4x^3+10x^2+16x+19+16x^(-1)+10x^(-2)+4x^(-3)+x^(-4)`