Question

What is the binomial expansion of `(a^2 + 2)^4`?

Answer 1

`color(red)((a^2+2)^4 = a^8 + 8a^6 + 24a^4 + 32a^2 + 16)`

Explanation:

Write out the fifth row of Pascal's triangle and make the appropriate substitutions.

Pascal's triangle is

(from www.kidshonduras.com)

The numbers in the fifth row are 1, 4, 6, 4, 1.

They are the coefficients of the terms in a fourth order polynomial.

`(x+y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4`

Your polynomial is `(a^2+2)^4`

Let `x=a^2` and `y=2`.

Then your polynomial becomes

`(a^2+2)^4= (x+y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4`

If we substitute the values for `x` and `y`, we get

`(a^2+2)^4 = (a^2)^4 + 4(a^2)^3(2) + 6(a^2)^2(2)^2 + 4(a^2)(2)^3 + 2^4`

`(a^2+2)^4 = a^8 + 8a^6 + 24a^4 + 32a^2 + 16`