# Polynomial

A **polynomial** is a function in one or more variables that consists of a sum of variables raised to nonnegative, integral powers and multiplied by coefficients from a predetermined set (usually the set of integers; rational, real or complex numbers; but in abstract algebra often an arbitrary field).

For example, these are polynomials:

- , in the variable
- , in the variables and
- , in the variable
- , in the variable

However,

are functions, but *not* polynomials, in the variable

## Introductory Topics

### A More Precise Definition

A polynomial in one variable is a function . Here, is the th coefficient and . 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 . For instance, and . When a polynomial is written in the form with , the integer 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 because . 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

where is a constant, the are (not necessarily distinct) complex numbers and 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 . 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. In quadratics roots are more complex and can simply be the square root of a prime number.

#### Factoring

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

This polynomial easily factors to:

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 . The Rational Root Theorem states that if has a rational root and this fraction is fully reduced, then is a divisor of and is a divisor of . 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 , 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 , or is less than that by a multiple of 2.

### Binomial Theorem

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

### Special Values

Given the coefficients of a polynomial, it is very easy to figure out the value of the polynomial on different inputs. In some cases, the reverse is also true. The most obvious example is also the simplest: for any polynomial , so the value of a polynomial at 0 is also the constant coefficient.

Similarly, , so the value at 1 is equal to the sum of the coefficients.

In fact, the value at any point gives us a linear equation in the coefficients of the polynomial. We can solve this system and find a unique solution when we have as many equations as we do coefficients. Thus, given the value of a polynomial and different points, we can always find the coefficients of the polynomial.