# Homework Help: Elementary math proof

1. Apr 30, 2016

### lolo94

1. The problem statement, all variables and given/known data
Disprove the following: There exists a polynomial f(x) with integer coefficients such that f(1) is even and f(3) is odd.

2. Relevant equations

3. The attempt at a solution
It's a little bit intuitive.
Proof
1 and 3 have the same parity. They are both odd
so if(odd)=odd then f(1)=odd and f(3)=odd
or if(odd)=even then f(1)=even and f(3)=even

is that right?

2. Apr 30, 2016

### Ray Vickson

No, it is not correct, because you are essentially assuming what you want to prove. In order to be able to assert that f(odd) = odd, for example, you need to show that it is not possible to have f(odd1) = odd but f(odd2) = even, etc.

3. Apr 30, 2016

### geoffrey159

Hint: if there was such a polynomial $f(3)-f(1)$ would be both even and odd.

4. Apr 30, 2016

### lolo94

why would they both be even and odd?

5. Apr 30, 2016

### Staff: Mentor

What do you know about $f(3) - f(1)$?

6. May 1, 2016

### haruspex

Let the polynomial be Σpnxn. What does f(3)-f(1) look like?