0
$\begingroup$

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?

Thanks a lot in advance...

SB7

$\endgroup$
1
$\begingroup$

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?

$\endgroup$
0
$\begingroup$

$\frac{1}{2}x^4-\frac{1}{2}x^2-\frac{1}{2}x+\frac{1}{2}$ has two roots in $[0,1]$ as well, there are many examples. (Unless I don't understand what you mean by integer degree).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.