What are common mistakes students make when using the fundamental theorem of algebra?
A few thoughts...
The number one mistake seems to be a mistaken expectation that the fundamental theorem of algebra (FTOA) will actually help you to find the roots that it tells you are there.
The FTOA tells you that any non-constant polynomial in one variable with complex (possibly real) coefficients has a complex (possibly real) zero.
A straightforward corollary of that, often stated with the FTOA, is that a polynomial in one variable with complex coefficients of degree
The FTOA does not tell you how to find the roots.
The very name "fundamental theorem of algebra" is something of a misnomer. It is not a theorem of algebra, but of analysis. It cannot be proved purely algebraically.
Another misunderstanding that could and probably does result from the FTOA is the belief that the complex numbers are unique in being algebraically closed in this way.
The smallest algebraically closed field containing the rational numbers