Question

Let p(x) be a polynomial with real coefficients such that p(n)=p(n-1) for any positive integer n. Prove that p(x)=p(0) for all x∈R.

Answer 1

This looks like a proof by induction. Start with n=1. You know that \(p(1)=p(1-1)=p(0)\). Assume it's true up to some integer \(k\), and show that it's true for \(k+1\).

Answer 2

but x may not be a integer.

Answer 3

Good point. Induction only takes care of the positive integers.

Answer 4

Is this the entire problem? If so, you could create a counterexample using the taylor expansion for \(\sin(x)\) such that it has a period of 1.

Answer 5

They might mean a finite polynomial. Weird things start happening when you consider infinite polynomials like taylor series.

Answer 6

very true. If it's finite, then this proposition is true.

Answer 7

thank you very much!

Answer 8

Did we do anything o.O lol

Answer 9

Not that I know of... does @Davidc still want help on proving this for finite degree polynomials?

Answer 10

no