# polynomial

#### jacks

##### Well-known member
A polynomial $f(x)$ has Integer Coefficients such that $f(0)$ and $f(1)$ are both odd numbers. prove that $f(x) = 0$ has no Integer solution

#### CaptainBlack

##### Well-known member
A polynomial $f(x)$ has Integer Coefficients such that $f(0)$ and $f(1)$ are both odd numbers. prove that $f(x) = 0$ has no Integer solution
There are details you may need to fill in yourself but:

$$f(0)$$ odd implies that the constant term is odd

then $$f(1)$$ odd implies that there are an even number of odd coeficients of the non constant terms.

So if $$x \in \mathbb{Z}$$ then $$f(x)$$ is odd and so cannot be a root of $$f(x)$$

CB