$P(x)=P(-x)$ holds for all values of $x$ ,two conditions I was doing a question on polynomials , where it was found that 
$P(x)=P(-x)$ in the interval $[-\sqrt2,\sqrt 2 ]$
It was then concluded that $P(x)=P(-x)$ holds for all values of $x$ ,"since it is a polynomial".
Can someone help me understand why it could be  generalized ?
Edit - $P(x)$ is a polynomial with real coefficients .

If polynomial $f(x)$ has degree less than $n$, then $n+1$ number of points on the polynomial determines the polynomial.
Since $P(x) - P(-x)$ is a polynomial, and there are infinite number of $t \in [-\sqrt 2, \sqrt2]$ satisfying $P(t) - P(-t) = 0$, we have $P(x) - P(-x) \equiv 0 $ as function. 
