Polynomials

Section 0.7 Polynomials

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 (4.5.27). 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.

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Subsection 0.7.1 Basic definitions

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Definition 0.7.1. Polynomials.

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Assume

X \subseteq C

a polynomial on

X

is a function

f

of the form

\begin{matrix} (0.7.1) & f (x) = \sum_{i = 1}^{\infty} a_{i} x^{i}, \end{matrix}

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where

a_{i} \in C

for all

i,

and there is a positive integer

n

such that

a_{i} = 0

for all

i > n .

Equivalently,

f

is a function of the form

\begin{matrix} (0.7.2) & f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} \dots + a_{1} x + a_{0}, \end{matrix}

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where

a_{i} \in C

for all

1 \leq i \leq n .

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We call

a_{i} x^{i}

the

i

-th term of

f,

and

a_{i}

the

i

-th coefficient;

a_{0}

is called the constant term of

f .

Furthermore, if in the expression (0.7.2) we have

a_{n} \neq 0,

then

a_{n} x^{n}

is called the leading term of

f,

and

a_{n}

its leading coefficient. Lastly, a root of

f

is a an element

c \in X

such that

f (c) = 0 .

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Remark 0.7.2. Polynomials as finite power series.

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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).

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Theorem 0.7.3. Basic properties of polynomials.

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Assume

X \subseteq C .

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If $f$ and $g$ are polynomials on $X,$ then so are $f + g,$ $f g,$ and $c f$ for any $c \in C .$
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If $f$ is a polynomial on $X$ and $c \in X$ is a root of $f,$ then there is a polynomial $g$ on $X$ such that

$\begin{matrix} (0.7.3) & f (x) = (x - c) g (x) . \end{matrix}$
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If $f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} \dots + a_{1} x + a_{0}$ is a nonzero polynomial on $X,$ then $f$ has at most $n$ distinct roots.

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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_{i} x^{i}

are uniquely determined by

f,

as long as the domain

X

is infinite. This furnishes us with a useful criterion for polynomial equality based on comparing coefficients.

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Corollary 0.7.4. Polynomial equality via coefficients.

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Assume

X

is an infinite subset of

C .

Let

f

and

g

be polynomials on

X

of the form

\begin{aligned} f (x) & = \sum_{i = 1}^{\infty} a_{i} x^{i} \\ g (x) & = \sum_{i = 1}^{\infty} b_{i} x^{i} . \end{aligned}

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We have $f = g$ if and only if $a_{i} = b_{i}$ for all $i .$
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In particular, $f$ is the zero function on $X$ if and only if $a_{i} = 0$ for all $i .$

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Subsection 0.7.2 Degree of a polynomial

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It follows from Corollary 0.7.4 that if

X

is an infinite subset of

C

and

f

is a nonzero polynomial on

X,

then

f

has a unique expression of the form

f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} \dots + a_{1} x + a_{0},

where

a_{n} \neq 0 .

The integer

n

appearing in this expression is an important invariant of

f

called its degree.

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Definition 0.7.5. Degree of a polynomial.

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Assume

X

is an infinite subset of

C

and let

f

be a nonzero polynomial on

X

of the form

f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} \dots + a_{1} x + a_{0}

with

a_{n} \neq 0 .

The integer

n

in this expression is called the degree of

f,

denoted

\deg f .

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We call the polynomial

f

linear if

\deg f = 1,

quadratic if

\deg f = 2,

and cubic if

\deg f = 3 .

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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.

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Theorem 0.7.6. Basic properties of degree.

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Assume

X

is an infinite subset of

C .

Let

f

and

g

be polynomials on

X .

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$\deg f g = \deg f + \deg g .$
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$\deg (f + g) \leq max {\deg f, \deg g} .$

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Subsection 0.7.3 Factoring polynomials

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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 : R \to R

that have no roots whatsoever: for example, from the quadratic formula we know that any quadratic polynomial

f (x) = a x^{2} + b x + c

satisfying

b^{2} - 4 a c < 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,

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.

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