I am confused with my knowledge of polynomials? Is $x^2+cos(x)=0$ a polynomial?

Please help( I know its a naive question)
It is not. If it were, then
$$\cos(x) = x^2 + \cos(x) - x^2$$ whould be a polynomial as well, bus $\cos(x)$ has too many zeros ( infinite). A (non-zero) polynomial can only have a finite number of zeros.
It's not! Only powers of $x$ can appear in a polynomial, no trig functions. Also, technically, a polynomial is an expression not an equality.