I am confused with my knowledge of polynomials?

Question

Is $x^2+cos(x)=0$ a polynomial?

Please help( I know its a naive question)

Answer 1

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.

Answer 2

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.

Answer 3

No, for two reasons:

1. If $f(x)= x^2 + \cos(x) $ were a polynomial, then some derivative of $f$ would be identically zero, but $f^{(n)}(x)=\pm \cos(x)$ or $\pm \sin(x)$ for $n>2$.

2. If $f(x)= x^2 + \cos(x) $ were a polynomial, then so would $\cos(x)$, but $\cos(x)$ is a bounded nonconstant function and the only bounded polynomials are the constant functions.