Polynomial

A polynomial 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 polynomials:

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

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

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 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 polynomial 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 polynomial in this form, because the roots will be $x_1,x_2,...,x_n$ . This also tells us that a polynomial can have up to $n$ distinct roots, where $n$ is its degree.