Question

How do you use the Binomial Theorem to expand `(1+x+x^2)^3`?

Answer 1

Use a variant of Pascal's triangle to find:

`(1+x+x^2)^3 = 1+3x+6x^2+7x^3+6x^4+3x^5+x^6`

Explanation:

This is a power of a trinomial, not a binomial so the binomial theorem does not help much.

However, note that `1, x, x^2` are in geometric progression, so we can use a variant of Pascal's triangle to find the coefficients we want. Each term in this variant is the sum of the three terms above it, rather than two...

`color(white)(0000000000)1`

`color(white)(0000000)1color(white)(00)1color(white)(00)1`

`color(white)(0000)1color(white)(00)2color(white)(00)3color(white)(00)2color(white)(00)1`

`color(white)(0)1color(white)(00)3color(white)(00)6color(white)(00)7color(white)(00)6color(white)(00)3color(white)(00)1`

Hence we find:

`(1+x+x^2)^3 = 1+3x+6x^2+7x^3+6x^4+3x^5+x^6`