$P(x)=P(-x)$ 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 .

Let $Q(x)=P(x)-P(-x)$, then $Q(x)$ is a real polynomial since $P(x)$ is (make sure you can show this!). By the assumption, $Q$ has infinitely many roots. But the only real polynomial with infinitely many roots is the zero polynomial. Hence $Q(x)=0$ for all real $x$, so $P(x)=P(-x)$ for all real $x$.
