**1 :**How many zeros (up to multiplicity) does the polynomial have

*over the complex numbers*?

There are zeros (up to multiplicity).

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

_

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).

Fundamental Theorem
of Algebra Any polynomial may be factored into a product of irreducible factors,
where those factors are, at most, degree one in the complex numbers. That is to say
for any of the following form; where each of (that is to say, each coefficient
is a real number), then we can
factor and rewrite it into the following form; Where and may be complex
numbers.

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 be a polynomial of degree with the form: Then the equation has at most
real solutions and *exactly* complex solutions *up to multiplicity* .

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

Number of solutions of a polynomialLet’s say we have the following polynomial; and
we want to determine how many values satisfy the equation . Currently our corollary
doesn’t quite work because it only applies when polynomials are equal to zero, but
we can rewrite our current polynomial and manipulate it into that form. Specifically;

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.