Question

A. Prove the polynomial identity for the cube of a binomial representing a difference.

(a - b)³=a³-3a²b+3ab²-b³

Answer 1

`(a-b)^3 = (a-b)^(a-b)^2\\\\=(a-b)(a^2 - 2ab +b^2)\\\\= a(a^2 -2ab +b^2) -b(a^2 - 2ab +b^2)\\\\= a^3 -2a^2b -ab^2 -a^2b + 2ab^2 -b^3\\\\= a^3 - 3a^2b + 3ab^2 -b^3`

Explanation:

I am not sure if you wanted it in exactly 4 steps, But this is the way to do it.

If in exactly 4 steps, omit the third step. Then we would get:

`(a-b)^3 = (a-b)^(a-b)^2\\\\=(a-b)(a^2 - 2ab +b^2)\\\\= a^3 -2a^2b -ab^2 -a^2b + 2ab^2 -b^3\\\\= a^3 - 3a^2b + 3ab^2 -b^3`

Hope that helps ya