Disproving a Polynomial with Integer Coefficients: Elementary Math Proof

In summary, the conversation discusses disproving the existence of a polynomial with integer coefficients that has an even value at 1 and an odd value at 3. The attempt at a solution states that if 1 and 3 have the same parity, then f(1) and f(3) must also have the same parity. However, this is not correct as it assumes what it is trying to prove. It is then hinted that if there was such a polynomial, f(3) - f(1) would be both even and odd, leading to a contradiction. The conversation ends with a question about the form of f(3) - f(1).
  • #1
lolo94
17
0

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?
 
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  • #2
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?

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
Hint: if there was such a polynomial ##f(3)-f(1)## would be both even and odd.
 
  • #4
geoffrey159 said:
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?
 
  • #5
lolo94 said:
why would they both be even and odd?
What do you know about ##f(3) - f(1)##?
 
  • #6
lolo94 said:
why would they both be even and odd?
Let the polynomial be Σpnxn. What does f(3)-f(1) look like?
 

What is a polynomial with integer coefficients?

A polynomial with integer coefficients is an algebraic expression that contains variables and coefficients that are all integers. For example, 3x^2 + 5x + 2 is a polynomial with integer coefficients.

Why is it important to disprove a polynomial with integer coefficients?

Disproving a polynomial with integer coefficients is important because it allows us to determine whether the polynomial has any solutions that are also integers. This can be useful in various mathematical applications and problem-solving.

What is an elementary math proof?

An elementary math proof is a mathematical argument that uses basic concepts and logical reasoning to prove a statement or theorem. It is typically a step-by-step process that follows established mathematical rules and principles.

How can we disprove a polynomial with integer coefficients?

To disprove a polynomial with integer coefficients, we can use the Rational Root Theorem or the Factor Theorem. These theorems allow us to test potential solutions and determine if they are valid roots of the polynomial. If no valid roots are found, the polynomial is disproven.

What are the limitations of disproving a polynomial with integer coefficients?

Disproving a polynomial with integer coefficients only works for polynomials with integer coefficients. It cannot be used for polynomials with non-integer coefficients or for more complex mathematical expressions. Additionally, it may not always be possible to find valid roots, so the polynomial may not be fully disproven.

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