If something holds for n+1 distinct values then it holds for all values.Proving a property of polynomial Ok I am stuck on proving a property of polynomial.It basically goes like this.
If $$f(x) = \sum_{k=0}^{n} c_kx^k $$ is equal to zero for n+1 distinct real values x,then f(x) is equal to 0 for all real x and $c_k = 0 $.
I have tried making a contradiction as in : There exist n+1 distinct real values of x for which f(x) = 0 and $c_k \neq 0 $ .
From there I tried proving a contradiction but I do not not how to proceed,or which polynomial to consider.If this contradiction is to be proven then it would follow that $c_k \neq 0 $ does not hold(because x values exist by hypothesis),and that would conclude it.
This problem comes from Tom Apostol Calculus vol.1 Chapter 1.5 exercise 9 ,subexercise d .
It would be much appreciated if you would offer your help,and if you would instruct me in how to go about proving things of this form(namely if some number of values is satisfy,then all values satisfy).
Thank you in advance  

You can also appeal to the mean value theorem. If $f$ has at least $n+1$ distinct zeroes, then $f'$ has at least $n$ distinct zeroes, $f''$ has at least $n-1$ distinct zeroes, and so on. Thus $f^{(n)}$ has at least one zero. Since $f^{(n)}$ is constant, you get $f^{(n)} \equiv 0$. Thus $f^{(n-1)}$ is constant, and since it has zeroes $f^{(n-1)} \equiv 0$. Keep going until you get $f \equiv 0$.
