Somebody can help me with this polynomial problem? I tried something but i am not really sure if i can finish with that. Thank you!
Let $P$ a polynomial with integer coefficients for which exists $2$ integer numbers, one odd, one even such that values of the polynomial in these values are odd. Show that the polynomial cannot have integer zeros.
I tried to use the contradiction and purpose we have zeros integer numbers. But i don't know how to elaborate that.
I used $P=a_nX^n+\cdots+a_0$ and $a,b\in Z,a=2k,b=2k+1,k\in Z$. Then we have $P(a)=2k+1$ and $P(b)=2k+1$. If we say $P$ has integer zeros let $a,b$ to be zeros. But actually $P(a)$ and $P(b)$ are even, contradiction?
Best Answer
Since $P(\text{some even})=\text{odd}$, the constant term must be odd, since the non-constant terms will always evaluate to an even number with an even argument. By the rational root theorem, then, any integer zero of $P$ must be odd.
Now consider $P(\text{some odd})=\text{odd}$. Any term with an even coefficient can be ignored, since they will not flip the parity of the result. The remaining terms, those with odd coefficients (including the constant term), will be odd no matter what odd number is used as the argument to $P$.
Since the result is odd, there must be an odd number of terms with odd coefficients, and since they stay odd for any odd argument, $P(\text{odd})=\text{odd}$ for all odd arguments. In particular, $P(\text{odd})\ne0$.
But we know that any integral root of $P$ must be odd. This is a contradiction. So $P$ has no integral roots.