# Polynomial proof

1. May 6, 2016

### lolo94

1. The problem statement, all variables and given/known data
Let f(x) = ax^2 + bx + c be a quadratic polynomial. Either prove or disprove the following statement: If f(0) and f(1) are even integers then f(n) is an integer for every natural number n.

2. Relevant equations

3. The attempt at a solution
I tried different approaches such as analyzing the constants, f(n)-f(0)-f(1).
How do you approach these problems in general?

2. May 6, 2016

### SammyS

Staff Emeritus
What can you tell about c from knowledge of f(0) ?

3. May 6, 2016

### lolo94

c=even integer

4. May 7, 2016

### Math_QED

We can construct a unique parabola using 3 points. Consider the function f$$x → ax^2 + bx + c$$

We know:
f(0) = a1
f(1) = a2

a1 and a2 are even integers. You can use f(n1) for the third point. Then you have a parabola through these 3 points. Try this.