This section covers one of the most important results in the last couple centuries in algebra; the so-called “Fundamental Theorem of Algebra.”
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.
In the last line above we have now manipulated our polynomial into a form where it equals zero, so our corollary tells us that there are at most real solutions (ie at most real values of that satisfies this equality) and exactly complex solutions. Remember that in both of these cases the “” is up to multiplicity.
With a little manipulation we can see explicitly what values of work;
Thus we see we have solutions of: with multiplicity of 2, (the larger of the two remaining zeros) and (the smaller of the two remaining zeros) both with multiplicity of 1. In this case, we do indeed have 4 solutions to the equation, but they are not unique as one of them has multiplicity of 2. That is to say, if we list the solutions from lowest (most negative) to largest (most positive) as: we have four solutions as the corollary claimed.