Bob tells me that the Identity Theorem is the three following statements
If a polynomial has infinitely many roots, then it is equal to $0$.
If two polynomials satisfy $P(x)=Q(x)$ for infinitely many $x$, then the two polynomials are equal.
If two polynomials of at most degree $n$ satisfy $P(x)=Q(x)$ for $n+1$ values of $x$, then the two polynomials are equal.
I do not know how to prove these; I think the Factor, Remainder, and Division Theorems will be useful.
I tried using Fundamental Theorem of Algebra, but it did not get me anywhere.
For polynomials with coefficients in $\mathbb C$, the three facts are easy consequences of the fundamental theorem of algebra, which says that a nonzero polynomial has $n$ ( not necessarily distinct) roots, where $n$ is the degree.
For example, $n+1$ would be too many roots, leaving the zero polynomial as the only way out.