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Section 0.7 Polynomials
Subsection Basic definitions
Definition 0.7.1 . Polynomials.
Assume \(X\subseteq \C\) a polynomial on \(X\) is a function \(f\) of the form
\begin{equation}
f(x)=\sum_{i=1}^\infty a_ix^i\text{,}\tag{0.7}
\end{equation}
where \(a_i\in \C\) for all \(i\text{,}\) and there is a positive integer \(n\) such that \(a_i=0\) for all \(i> n\text{.}\) Equivalently, \(f\) is a function of the form
\begin{equation}
f(x)=\anpoly\text{,}\tag{0.8}
\end{equation}
where \(a_i\in \C\) for all \(1\leq i\leq n\text{.}\)
We call
\(a_ix^i\) the
\(i\) -th
term of
\(f\text{,}\) and
\(a_i\) the
\(i\) -th
coefficient ;
\(a_0\) is called the
constant term of
\(f\text{.}\) Furthermore, if in the expression
(0.8) we have
\(a_n\ne 0\text{,}\) then
\(a_nx^{n}\) is called the
leading term of
\(f\text{,}\) and
\(a_{n}\) its
leading coefficient . Lastly, a
root of
\(f\) is a an element
\(c\in X\) such that
\(f(c)=0\text{.}\)
Theorem 0.7.3 . Basic properties of polynomials.
Assume \(X\subseteq \C\text{.}\)
If
\(f\) and
\(g\) are polynomials on
\(X\text{,}\) then so are
\(f+g\text{,}\) \(fg\text{,}\) and
\(cf\) for any
\(c\in \C\text{.}\)
If \(f\) is a polynomial on \(X\) and \(c\in X\) is a root of \(f\text{,}\) then there is a polynomial \(g\) on \(X\) such that
\begin{equation}
f(x)=(x-c)g(x)\text{.}\tag{0.9}
\end{equation}
If
\(f(x)=\anpoly\) is a nonzero polynomial on
\(X\text{,}\) then
\(f\) has at most
\(n\) distinct roots.
An important consequence of statement (
3 ) of
Theorem 0.7.3 is that the coefficients
\(a_i\) of a polynomial
\(f(x)=\sum_{i=1}^\infty a_ix^i\) are uniquely determined by
\(f\text{,}\) as long as the domain
\(X\) is
infinite . This furnishes us with a useful criterion for polynomial equality based on comparing coefficients.
Corollary 0.7.4 . Polynomial equality via coefficients.
Assume \(X\) is an infinite subset of \(\C\text{.}\) Let \(f\) and \(g\) be polynomials on \(X\) of the form
\begin{align*}
f(x) \amp =\sum_{i=1}^\infty a_ix^i \\
g(x)\amp =\sum_{i=1}^\infty b_ix^i\text{.}
\end{align*}
We have
\(f=g\) if and only if
\(a_i=b_i\) for all
\(i\text{.}\)
In particular,
\(f\) is the zero function on
\(X\) if and only if
\(a_i=0\) for all
\(i\text{.}\)
Subsection Degree of a polynomial
It follows from
Corollary 0.7.4 that if
\(X\) is an infinite subset of
\(\C\) and
\(f\) is a nonzero polynomial on
\(X\text{,}\) then
\(f\) has a unique expression of the form
\(f(x)=\anpoly\text{,}\) where
\(a_n\ne 0\text{.}\) The integer
\(n\) appearing in this expression is an important invariant of
\(f\) called its
degree .
Definition 0.7.5 . Degree of a polynomial.
Assume
\(X\) is an infinite subset of
\(\C\) and let
\(f\) be a nonzero polynomial on
\(X\) of the form
\(f(x)=\anpoly\) with
\(a_n\ne 0\text{.}\) The integer
\(n\) in this expression is called the
degree of
\(f\text{,}\) denoted
\(\deg f\text{.}\)
We call the polynomial
\(f\) linear if
\(\deg f=1\text{,}\) quadratic if
\(\deg f=2\text{,}\) and
cubic if
\(\deg f=3\text{.}\)
Defining the degree of the zero function to be
\(-\infty\) may seem a little peculiar. Do not be disturbed! You can think of this as a convention that allows for clean statements of theorems like the following.
Theorem 0.7.6 . Basic properties of degree.
Assume
\(X\) is an infinite subset of
\(\C\text{.}\) Let
\(f\) and
\(g\) be polynomials on
\(X\text{.}\)
\(\deg fg=\deg f+\deg g\text{.}\)
\(\deg (f+g)\leq \max\{\deg f, \deg g\}\text{.}\)
Subsection Factoring polynomials
Statement (2) of
Theorem 0.7.3 makes a connection between roots of a polynomial and factorization; and the expression
(0.9) can be thought of as a first step in writing the polynomial
\(f\) as a product of linear polynomials. The existence of roots depends on the given domain of the function. Indeed, there are plenty of polynomials
\(f\colon \R\rightarrow \R\) that have no roots whatsoever: for example, from the quadratic formula we know that any quadratic polynomial
\(f(x)=ax^2+bx+c\) satisfying
\(b^2-4ac\lt 0\) has no real roots. According to the
fundamental theorem of algebra , the situation over
\(\C\) is vastly different: not only does every polymonial have a root in
\(\C\text{,}\) we can factor it completely as a product of linear polynomials. This is one of the main incentives for introducing the complex numbers as a number system.
Theorem 0.7.7 . Fundamental theorem of algebra.
Any polynomial \(f(x)=\anpoly\text{,}\) thought of as a function from \(\C\) to itself, can be factored into linear terms as
\begin{equation*}
f(x)=(x-z_1)(x-z_2)\cdots (x-z_n)\text{,}
\end{equation*}
where the \(z_i\) are (not necessarily distinct) complex numbers.
Exercises Exercises
1. 2.
