Is There A Polynomial That Has Infinitely Many Roots? Is there a polynomial function $P(x)$ with real coefficients that has an infinite number of roots? What about if $P(x)$ is the null polynomial, $P(x)=0$ for all x?

By Gauss's fundamental theorem of algebra a polynomial has number of roots equal to its degree, where roots are counted with multiplicities. So in order to have infinite roots we should consider asymptotically infinite degree polynomial. 
Let us consider a three term recurrence,  $$ P_{n+1}(z) = (z^3-z)P_{n}(z)-P_{n-1}(z).$$ With initial conditions $$P_0(z) = 1 $$ and $$P_1(z) = z^3-z+1.$$ 
As $$ \lim_{n-> \infty}  P_{n+1}(z) $$ will have infinite roots. Note that all the coefficients of this polynomial are real. Asymptotically the roots fall on the projection of the polynomial  $$ z^3-z = r $$ on complex plane. Here $$r \in (-2i, 2i)$$ and $z$ is any complex number. We can observe this in case of spectrum of tridiagonal k Toeplitz matrices. 
