Elementary math proof

  • Thread starter lolo94
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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?
 

Answers and Replies

  • #2
Ray Vickson
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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.
 
  • #3
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Hint: if there was such a polynomial ##f(3)-f(1)## would be both even and odd.
 
  • #4
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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?
 
  • #6
haruspex
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why would they both be even and odd?
Let the polynomial be Σpnxn. What does f(3)-f(1) look like?
 

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