The way you have written it, with y constant, it is not true. (Hence chiro's question.) Rather you need something like
That is, you need that "y" different for each power of x and, obviously, if you are multiplying two nth degree polynomials together, you do NOT get an nthe degree polynomial, you get a 2n degree polynomial:
(a+ bx)(c+ dx)= ac+ (ad+ bc)x+ (bd)x^2.
And...I know i don't get a nth degree polynomial... I edited some time ago to change the picture and replace the n with infinite (I replaced it yesterday)...
And the only thing I have to prove is that the result is a sum of the same form.
This, correcting the with
Again, that certainly is NOT true! Unless you mean, as both chiro and I said before, you mean .
And, in that case, it is pretty close to trivial. The product of any two terms, times must be of the form so the only thing you can have are "numbers times non-negative powers of x" and that is all the term " means!