# I need to prove a simple proposition about polynomials

I am working on systems theory and I have reduced one problem that I have into a simple polynomial proposition.

All (non-constant) polynomials (with integer degrees) in which the sum of positive coefficients is exactly 1 and the sum of negative coefficients is in [-1,0] have maximum one solution in the range [0-1].

Can somebody either help me or find one polynomial that disproves the proposition?

Hint For $c \neq 0$, the polynomial $$c \left(x - \tfrac{1}{3}\right) \left(x - \tfrac{2}{3}\right) = c \left(x^2 - x + \frac{2}{9}\right)$$ has two roots in $[0, 1]$. Can you choose $c$ so that the polynomial is a counterexample?
$\frac{1}{2}x^4-\frac{1}{2}x^2-\frac{1}{2}x+\frac{1}{2}$ has two roots in $[0,1]$ as well, there are many examples. (Unless I don't understand what you mean by integer degree).