What Are Polynomial Functions?

A polynomial is an expression where all the terms have $x$ as the base, and where the exponents are non-negative integers (the exponents can be 0, 1, 2, $\dots$ ). The largest exponent of the expression determines the degree of the polynomial.

Theory

Polynomials

A polynomial of degree $n$ looks like this:

\begin{array}{l} a_{n} x^{n} + a_{n - 1} x^{n - 1} + a_{n - 2} x^{n - 2} + \dots \\ + a_{2} x^{2} + a_{1} x + a_{0} \end{array}

\begin{array}{l} a_{n} x^{n} + a_{n - 1} x^{n - 1} + a_{n - 2} x^{n - 2} + \dots + a_{2} x^{2} + a_{1} x + a_{0} \end{array}

Note! Remember that constants, which don’t appear at first glance to have an $x$ in them, actually have an $x$ raised to the power zero – $x^{0}$ – which equals $1$ . A polynomial function is a function where $f (x)$ is set equal to a polynomial. Linear functions and quadratic functions are the most common polynomial functions. Linear functions are degree 1 polynomials and quadratic functions are degree 2 polynomials. You may also encounter polynomial functions of a higher degree than this. Below, you can see a picture of five different polynomial functions with degrees from 2 to 7. For example, the pink graph is a quadratic function and the gray is a function of third degree.

Graph of five polynomials plotted in the same coordinate system

One can usually see what degree a polynomial has by looking at its graph. A quadratic polynomial has one “bend”, a cubic (third degree) polynomial has two “bends”, a quartic (fourth degree) polynomial has three “bends”, etc.

But note that in some special cases, these “bends” are at the same point—and then you won’t see all of them.

Example 1

$f (x) = 2 x^{3} - 4 x + 3$ is a polynomial of degree 3 (a cubic polynomial).

$f (x) = 4 x^{2} - 3 x^{3} + x^{5}$ is a polynomial of degree 5 (a quintic polynomial).

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What Are Quadratic Functions?

What Are Exponential Functions?

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