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?
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.
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.
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.
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.
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.