system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: The intuition behind the fundamental theorem of algebra

In algebra, we learn that a polynomial of degree n must cross the x-axis n times. But not every polynomial seems to behave this way--some cross fewer times than their degrees. As a solution, we are told the function also crosses through complex roots on an imaginary axis to account for those missing roots.

So my question is: how did they come up with the fundamental theorem if they hadn't known about complex roots yet? If not every polynomial seemed to satisfy the theorem, it shouldn't have even been created in the first place. It would have just been assumed that some polynomials have fewer roots than their degrees.

The paradox is that the claims of the theorm served as the driving force to discover complex numbers, even though complex numbers are what created the intuition behind the theorem in the first place.
gpt
I think you're a little mixed up, it's not that a nth degree polynomial crosses the x axis n times, it's that the polynomial has n roots, some of which may or may not be imaginary. But then certian polynomials had roots that didn't seem to exist. For instance if you have x^2 = -1 as a root you can't really solve this because if you multiply a number by itself it should always be positive. So mathematicians started saying that the solution to this root is an imaginary number. For about 200 years this imaginary number had no real use except to factor these polynomials, that is until later mathematicians such as Cauchy came along and showed that there are ways to visualize imaginary numbers as complex numbers in a complex plane.