The discussion centers on disproving the existence of a polynomial f(x) with integer coefficients such that f(1) is even and f(3) is odd. The key argument presented is that both 1 and 3 share the same parity, leading to the conclusion that if f(1) is even, then f(3) must also be even, contradicting the assumption. The proof hinges on the evaluation of the polynomial at these points and the implications of their parities, specifically that f(3) - f(1) cannot simultaneously be even and odd.
PREREQUISITESMathematics students, educators, and anyone interested in number theory or polynomial functions will benefit from this discussion.
lolo94 said: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?
why would they both be even and odd?geoffrey159 said:Hint: if there was such a polynomial ##f(3)-f(1)## would be both even and odd.
What do you know about ##f(3) - f(1)##?lolo94 said:why would they both be even and odd?
Let the polynomial be Σpnxn. What does f(3)-f(1) look like?lolo94 said:why would they both be even and odd?