Given a polynomial p, prove A^2-B^2 = p(1)p(-1)

[Robb]

Homework Statement

Given a polynomial p, let A be the sum of the coefficients of the even powers, and let B be the sum of the coefficients of the odd powers. Prove that A^2 - B^2 = p(1)p(-1).

Homework Equations

The Attempt at a Solution

See attached. Can someone please look at my work to see if it is correct. Best!

 

[kuruman]

Your solution is correct but limited. Can you show this to be true for an arbitrary number of terms $N$ without going through all the tediousness of your method? Generous hint: Can you express A and B in terms of P(1) and P(-1)?

 

[Robb]

Well that was a lot easier! Thanks!