Hello, here is the question I am working on:

If u and v are two roots of the polynomial

then show u *v is a root of

Now I know that this is true, and the brute force is to consider that the product of the roots of the first polynomial is -1, then match up the expansion of the second polynomial. I was wondering if there is a special relation between the roots of a sum of polynomials to the roots of the originals, since the second polynomial is the sum of the first one and

which has simple roots -1,1,0,0,-i,i

Does anyone know a shortcut proof for this?

Thanks.