Polynomials will play an important role in our treatment of linear algebra both in terms of building examples of vector spaces (3.3.20) and as tools for studying linear operators (5.5.27). We gather here some basic definitions and properties of polynomials. Although we will primarily be concerned with polynomials with real coefficients in this course, it is convenient to develop the theory over the complex numbers. As with Section 0.6, all theorems will be stated without proof.
Subsection0.7.1Basic definitions
Definition0.7.1.Polynomials.
Assume \(X\subseteq \C\) a polynomial on \(X\) is a function \(f\) of the form
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
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 (0.7.2) 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 \(f(c)=0\text{.}\)
Remark0.7.2.Polynomials as finite power series.
As we see in the expression (0.7.1) we have essentially defined a polynomial to be a function with a finite power series representation. This may seem somewhat overkill in terms of the level of abstraction, but makes for simpler exposition in the discussion below. In any case you are always welcome to revert to the more familiar characterization in (0.7.2).
Theorem0.7.3.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
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.
Corollary0.7.4.Polynomial equality via coefficients.
Assume \(X\) is an infinite subset of \(\C\text{.}\) Let \(f\) and \(g\) be polynomials on \(X\) of the form
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 \(i\text{.}\)
Subsection0.7.2Degree 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.
Definition0.7.5.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.
Theorem0.7.6.Basic properties of degree.
Assume \(X\) is an infinite subset of \(\C\text{.}\) Let \(f\) and \(g\) be polynomials on \(X\text{.}\)
\(\deg fg=\deg f+\deg g\text{.}\)
\(\deg (f+g)\leq \max\{\deg f, \deg g\}\text{.}\)
Subsection0.7.3Factoring polynomials
Statement (2) of Theorem 0.7.3 makes a connection between roots of a polynomial and factorization; and the expression (0.7.3) 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.
Theorem0.7.7.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
where the \(z_i\) are (not necessarily distinct) complex numbers.
Exercises0.7.4Exercises
1.
Find a degree 3 polynomial whose coefficient of \(x^3\) equal to 1. The zeros of this polynomial are \(3\text{,}\)\(-4 i\text{,}\) and \(4 i\text{.}\) Simplify your answer so that it has only real numbers as coefficients.
