Introduction

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Section 7.7 Introduction

Definition 7.7.1.

For \(n\in\NN\text{,}\) an \(n\)-degree complex polynomial \(p\) is a function of the form

\begin{equation} p(z)=a_nz^n + a_{n-1}z^{n-1} + \cdots + a_0\label{e_pz}\tag{7.7.1} \end{equation}

where \(a_n \neq 0\) and \(a_i \in \CC\) for all \(i\text{.}\)

\(n\) is called the degree of the polynomial.

A root of \(p\) is a complex number \(\alpha\in\CC\) such that \(p(\alpha)=0\text{.}\)

If the degree of a polynomial is \(2\) then the polynomial is called a quadratic. Any quadratic \(az^2+bz+c\) has two roots which can be found using the familiar quadratic formula:

\begin{equation*} \frac{-b\pm \sqrt{b^2-4ac}}{2a}\text{.} \end{equation*}

Note: One needs to be careful here, since we are allowing \(a,b,c\in\CC\text{,}\) so \(b^2-4ac\) could be complex. In this case, \(\sqrt{b^2-4ac}\) is interpreted to mean any number \(z\) so that \(z^2=b^2-4ac\text{,}\) and then the solutions are \(\frac{-b\pm z}{2a}\text{.}\)

There are also formulae for the roots of cubic or quartic (i.e. degree 3 or 4) polynomials, although these are much less convenient to write down. It is natural to ask whether there are convenient formulae for higher order polynomials, but this is not the case:

Theorem 7.7.2. Abel-Ruffini Theorem.

There is no formula (like the quadratic formula) for the roots of a polynomial of degree \(\geq 5\text{.}\)

This doesn't mean degree \(5\) polynomials don't have roots, we will see below that they must have at least one root. This also doesn't mean it is impossible to solve higher order polynomials for their roots, it just means we do not have an expression (i.e. formula) for the roots of the polynomial in terms of the coefficients.