# Difference between revisions of "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:

• $4x^2 + 6x - 9$, 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,

• $\sin^2{x} + 5$
• $\frac{4x+3}{2x-9}$
• $x^{-1}+2+3x+x^2$
• $x^{1/3}=\sqrt[3]{x}$

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 $\deg(P)$. 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. 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:

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

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 $P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0$. The Rational Root Theorem states that if $P(x)$ has a rational root $\pm\frac{p}{q}$ and this fraction is fully reduced, then $p$ is a divisor of $a_0$ and $q$ is a divisor 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.

### 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 $P(x) = a_nx^n + \ldots + a_1 x + a_0$, $P(0) = a_0$ so the value of a polynomial at 0 is also the constant coefficient.

Similarly, $P(1) = a_n + a_{n - 1} + \ldots + a_1 + a_0$, 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 $P$ and $\deg(P) + 1$ different points, we can always find the coefficients of the polynomial.