Question to ponder about - - # of nonzero integer coefficients of a polynomial squared

[checkitagain]

Suppose you look at polynomials, P(x), of degree n, with all nonzero integer coefficients
and, in particular, a coefficient of 1 for the nth degree (leading) term. And look at those
polynomials whose squares have the fewest number of nonzero integer coefficients
possible.
Examples:
----------[tex](x + 1)^2 \ = \ x^2 + 2x + 1 \ \ \ \ --> \ 3 \ \ terms[/tex][tex](x^2 + 2x - 2)^2 \ = \ x^4 + 4x^3 - 8x + 4 \ \ \ \ --> \ 4 \ \ terms[/tex]
[tex](x^3 + x^2 + 2x - 2)^2 \ = \ x^6 + 2x^5 + 5x^4 - 8x + 4 \ \ \ \ --> \ 5 \ \ terms[/tex]

[tex](x^3 + 2x^2 - 2x - 1)^2 \ = \ x^6 + 4x^5 - 10x^3 + 4x + 1 \ \ \ \ --> \ 5 \ \ terms[/tex]

[tex](x^3 + 2x^2 - 2x + 4)^2 \ = \ x^6 + 4x^5 + 20x^2 - 16x + 16 \ \ \ \ --> \ 5 \ terms[/tex]
[tex](x^4 + 2x^3 - 2x^2 + 4x + 4)^2 \ = \ x^8 + 4x^7 + 28x^4 + 32x + 16 \ \ \ \ --> \ 5 \ terms[/tex]__________________________________________________
Can the minimum number of nonzero integer coefficients possible
for the squares of polynomial P(x) be less than the degree of that P(x)?** All polynomials are taken to be simplified, including having all
like terms combined.

 

[Aaron Bex]

No, the minimum number of nonzero integer coefficients possible for the squares of polynomial P(x) cannot be less than the degree of that P(x). This is because the square of a polynomial will always have an exponent of at least double the degree of the original polynomial. For example, if P(x) is a polynomial of degree 3, then P(x)^2 will have an exponent of 6. Thus, there must be at least 6 nonzero integer coefficients for P(x)^2.