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This section covers one of the most important results in the last couple centuries in algebra; the so-called “Fundamental Theorem of Algebra.”

We will spend considerable time learning how to manipulate polynomials, typically in an effort to ascertain certain properties or values. It helps then, to know that our goal of factoring/manipulating the polynomials is actually possible. This is what the Fundamental Theorem of Algebra (and its corollaries) gives us, which we explore below.
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##### Fundamental Theorem of Algebra, aka Gauss makes everyone look bad.

In grade school, many of you likely learned some variant of a theorem that says any polynomial can be factored to be a product of smaller polynomials; specifically polynomials of degree one or two (depending on your math book/teacher they may have specified that they are polynomials of degree one, or so-called ‘linear’ polynomials). This is a paraphrased version of a very important and surprisingly recent theorem in mathematics called The fundamental theorem of algebra which is recorded formally here (don’t worry, explanation will follow).

In essence, what this theorem is saying, is that each polynomial can be factored down to the product of smaller polynomials, and that those polynomials can be made to be at most degree one if we allow complex numbers. This has a corollary,

specifically;

Let’s see this corollary in action in an example.

1 : How many zeros (up to multiplicity) does the polynomial $p(x) = 3x^2 - 2x^3 + x$ have over the complex numbers?
There are $\answer {3}$ zeros (up to multiplicity).