Polynomial

A polynomial is a function in one or more variables that consists of a sum of variables raised to integral powers and multiplied by coefficients.

For example, these are polynomials:

$4x^2 + 6x - 9$ , in the variable $x$
$x^3 + 3x^2y + 3xy^2 + y^3$ , in the variables $x$ and $y$
$5x^4 - 2x^2 + 9$ , in the variable $x$
$\sin^2{x} + 5$ , in the variable $\sin x$

However,

$\sin^2{x} + 5$
$\frac{4x+3}{2x-9}$

are functions, but not polynomials, in the variable $x$

Introductory Topics

A More Precise Definition

A polynomial in one variable is a function $P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0$ . Here, $a_i$ is the $i$ th coefficient and $a_n \neq 0$ . Often, the leading coefficient of a polynomial will be equal to 1. In this case, we say we have a monic polynomial.

The Degree of a Polynomial

The simplest piece of information that one can have about a polynomial of one variable is the highest power of the variable which appears in the polynomial. This number is known as the degree of the polynomial and is written $\displaystyle \deg(P)$ . For instance, $\deg(x^2 + 3x + 4) = 2$ and $\deg(x^5 - 1) = 5$ . When a polynomial is written in the form $P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0$ with $a_n \neq 0$ , the integer $n$ is the degree of the polynomial.

The degree, together with the coefficient of the largest term, provides a surprisingly large amount of information about the polynomial: how it behaves in the limit as the variable grows very large (either in the positive or negative direction) and how many roots it has.

Finding Roots of Polynomials

What is a root?

A root is a value for a variable that will make the polynomial equal zero. For an example, 2 is a root of $x^2 - 4$ because $2^2 - 4 = 0$ . For some polynomials, you can easily set the polynomial equal to zero and solve or otherwise find roots, but in some cases it is much more complicated.

The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that any polynomial with complex coefficients can be written as

$P(x) = k(x-x_1)(x-x_2)\cdots(x-x_n)$ where $k$ is a constant, the $x_i$ are (not necessarily distinct) complex numbers and $n$ is the degree of the polynomial in exactly one way (not counting re-arrangements of the terms of the product). It's very easy to find the roots of a polynomial in this form because the roots will be $x_1,x_2,...,x_n$ . This also tells us that the degree of a given polynomial is at least as large as the number of distinct roots of that polynomial.

Factoring

Different methods of factoring can help find roots of polynomials. Consider this polynomial:

$x^3 + 3x^2 - 4x - 12 = 0$

This polynomial easily factors to:

$(x+3)(x^2-4) = 0$

$(x+3)(x-2)(x+2) = 0$

Now, the roots of the polynomial are clearly -3, -2, and 2.

The Rational Root Theorem

We are often interested in finding the roots of polynomials with integral coefficients. Consider such a polynomial $P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0$ . It can be shown that if $P(x)$ has a rational root $\pm\frac{p}{q}$ and this fraction is fully reduced, then $p$ is a factor of $a_0$ and $q$ is a factor of $a_n$ . This is convenient because it means we must check only a small number of cases to find all rational roots of many polynomials. It is also especially convenient when dealing with monic polynomials.

Descartes' Law of Signs

By the Fundamental Theorem of Algebra, the maximum number of distinct factors (not all necessarily real) of a polynomial of degree n is n. This tells us nothing about whether or not these roots are positive or negative. Decartes' Rule of Signs says that for a polynomial P(x), the number of positive roots to the equation is equal to the number of sign changes in the coefficients of the polynomial, or is less than that number by a multiple of 2. The number of negative roots to the equation is the number of sign changes in the coefficients of P(-x), or is less than that by a multiple of 2.

Binomial Theorem

The Binomial Theorem can be very useful for factoring and expanding polynomials.

Intermediate Topics

Olympiad Topics

Other Resources

An extensive coverage of this topic is given in A Few Elementary Properties of Polynomials by Adeel Khan.

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