Polynomial

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

For example, these are polynomials:

$4x^2 + 6x - 9$
$x^3 + 3x^2y + 3xy^2 + y^3$
$5x^4 - 2x^2 + 9$

These aren't polynomails:

$\sin^2{x} + 5$
$(4x+3)/(2x-9)$

Introductory Topics

A More Precise Definition

A polynomail 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, $a_n$ is the $n$ th coefficient, and $n$ is an integer.

Finding Roots of Polynomails

What is a root?

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

The Fundamental Theorem of Algebra

The fundamental theorem of algebra states that any polynomail can be written as:

$P(x) = k(x-x_1)(x-x_2)\cdots(x-x_n)$ where $k$ is a constant, and $n$ is the highest power of $x$ that $P(x)$ contains (also called the degree). It's very easy to find the roots of a polynomail in this form, because the roots will be $x_1,x_2,...,x_n$ . This also tells us that a polynomail can have up to $n$ distinct roots, where $n$ is its degree.