Question

How do you find the number of roots for `f(x) = x^4 + 2x^3 - 3x^2 - 4x - 12` using the fundamental theorem of algebra?

Answer 1

`f(x)` is of degree `4` so has exactly `4` zeros counting multiplicity.

Explanation:

The Fundamental Theorem of Algebra states that every non-constant polynomial equation in one variable with Complex coefficients has a Complex root.

Complex numbers here include Real numbers, that is numbers of the form `a+bi` with `b=0`.

A simple corollary of this, sometimes stated as part of the fundamental theorem is that a polynomial of degree `n > 0` has exactly `n` Complex roots counting multiplicity.

To prove the corollary from the theorem, find the first root then divide your polynomial by the corresponding factor. The resulting quotient polynomial will have degree `n-1`. If that is still non-zero, apply the theorem to know that there's another root to find, etc. Note that some of the roots found may be the same as one another, which is why we need to count multiplicity.