Disproving a Polynomial with Integer Coefficients: Elementary Math Proof

[lolo94]

Homework Statement

Disprove the following: There exists a polynomial f(x) with integer coefficients such that f(1) is even and f(3) is odd.

Homework Equations

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?

 

[Ray Vickson]

Homework Statement

Disprove the following: There exists a polynomial f(x) with integer coefficients such that f(1) is even and f(3) is odd.

Homework Equations

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?

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.

 

[geoffrey159]

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

 

[lolo94]

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

why would they both be even and odd?

 

[fresh_42]

why would they both be even and odd?

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

 

[haruspex]

why would they both be even and odd?

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