Polynomial Functions

Table of Contents

Polynomial functions are a central object in algebra. They generalize the polynomials you met in Algebra I (expressions like $2x^2 - 3x + 5$) to full-fledged functions: rules that take in a number $x$ and output another number.

In this chapter we focus on what makes a function polynomial, how to recognize one, and the basic features that distinguish polynomial functions from other kinds of functions. More detailed topics such as degree and long-term behavior, and the zeros of polynomial functions, will be developed in their own sections.

What Is a Polynomial Function?

A polynomial function in one variable $x$ is any function that can be written in the form
$$
f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_2 x^2 + a_1 x + a_0,
$$
where

$n$ is a nonnegative integer (that is, $n = 0, 1, 2, \dots$),
the numbers $a_0, a_1, \dots, a_n$ are real numbers called coefficients,
$a_n \neq 0$ (so that $n$ is really the highest power that appears).

The function $f$ takes an input $x$ and produces a value by plugging $x$ into this expression.

Key restrictions:

The exponent on $x$ must be a whole number (nonnegative integer: $0,1,2,\dots$).
No variable may appear in a denominator.
No variable may appear inside a root, exponent, logarithm, trigonometric function, and so on.

Polynomial functions in more than one variable (like $f(x,y) = 2x^2y + 3y^3$) exist, but here we concentrate on a single variable.

Recognizing Polynomial Functions

The most important skill at this stage is distinguishing polynomial functions from non-polynomial ones.

Acceptable Terms

Each term of a one-variable polynomial function looks like
$$a_k x^k$$
with:

$a_k$ a real number (which might be $0$), and
$k$ a nonnegative integer.

Examples of terms that are allowed:

$5x^3$
$-\tfrac{1}{2}x^2$
$7x$
$4$ (this is $4x^0$)
$0.03x^{10}$

A polynomial function is a sum of such allowed terms.

Non-Examples: What Disqualifies a Function from Being Polynomial?

Any of these features makes a function not polynomial in $x$:

Negative exponents

Example: $f(x) = 3x^{-1} + 2$ (same as $3/x + 2$)
The term $x^{-1}$ has a negative exponent, so this is not a polynomial function.

Fractional exponents

Example: $g(x) = x^{1/2} + 4$ (same as $\sqrt{x} + 4$)
Exponent $1/2$ is not a nonnegative integer, so this is not a polynomial function.

Variable in the denominator

Example: $h(x) = \dfrac{2x + 1}{x}$.
Even though the numerator is polynomial, $x$ appears in the denominator, so $h$ is not a polynomial function.

Variable inside a root

Example: $p(x) = \sqrt{x^2 + 1}$.
The variable expression is inside a square root; this is not a polynomial function.

Variable in an exponent (exponential function)

Example: $q(x) = 2^x + 3$.
Here $x$ is in the exponent, not the base, so this is not polynomial.

Variable inside other special functions

Example: $r(x) = \sin x$, $s(x) = \ln x$, etc.
Trigonometric, logarithmic, and similar functions are not polynomial functions.

Practice in Classification (Conceptual)

For each of the following, decide whether it is a polynomial function of $x$ and, if so, why:

$f(x) = x^4 - 2x^2 + x - 7$

All exponents are nonnegative integers; sum of allowed terms → polynomial.

$g(x) = 3x^3 + \dfrac{1}{x}$

Contains $x^{-1}$ (via $1/x$); negative exponent → not polynomial.

$h(x) = 5 - 2\sqrt{x}$

Contains $x^{1/2}$; fractional exponent → not polynomial.

$k(x) = \dfrac{1}{2}x^5 - 7x + 9$

All exponents $5, 1, 0$ are nonnegative integers; coefficients are real numbers → polynomial.

When recognizing a polynomial, always scan:

the exponents on $x$,
whether $x$ appears in denominators, roots, or exponents of other functions.

Constant and Zero Polynomial Functions

Some polynomial functions look extremely simple but are important to understand clearly.

Constant Polynomial Functions

A constant polynomial function has the form
$$
f(x) = c,
$$
where $c$ is a real number and does not depend on $x$.

Examples:

$f(x) = 5$
$g(x) = -\dfrac{3}{2}$
$h(x) = 0.01$

These are polynomial functions because they can be written as $c x^0$ with exponent $0$.

The Zero Polynomial Function

The function
$$
f(x) = 0
$$
for all $x$ is called the zero polynomial function.

It is special because all coefficients are $0$.
It does not behave like other polynomials when you talk about degree or zeros; those subtleties are handled in the later section on degree and behavior.

For now, the key point is: $f(x) = 0$ is indeed a polynomial function.

Standard Form and Like Terms

A polynomial can often be written in many different ways. For polynomial functions, it is convenient to use standard form.

A polynomial in $x$ is in standard form when it is written as
$$
a_n x^n + a_{n-1} x^{n-1} + \dots + a_2 x^2 + a_1 x + a_0,
$$
with powers of $x$ in descending order from highest exponent to lowest.

Examples of rewriting into standard form:

Start with
$$
f(x) = 2 - 3x^2 + x^4 + x.
$$

Highest power is $x^4$, then $x^2$, then $x$, then constant.
Standard form:
$$
f(x) = x^4 - 3x^2 + x + 2.
$$

Start with
$$
g(x) = 5x - 2x^3 + 7 - x.
$$

First, combine like terms ($5x$ and $-x$):
$$
g(x) = -2x^3 + 4x + 7.
$$
Now it is already in standard form: powers are $3, 1, 0$.

Like terms are terms that have exactly the same variable part (same variable to the same power), such as $5x^2$ and $-3x^2$. Constant terms (like $2$ and $-7$) are also like terms with each other.

Rewriting into standard form is especially useful later when discussing the degree and leading term of a polynomial function.

Basic Evaluation of Polynomial Functions

To evaluate a polynomial function at a specific input, you substitute the value for $x$ and follow the order of operations.

Example:
Let
$$
f(x) = 2x^3 - 3x + 1.
$$
Evaluate $f(2)$.
Substitute $x = 2$:
$$
f(2) = 2(2)^3 - 3(2) + 1 = 2 \cdot 8 - 6 + 1 = 16 - 6 + 1 = 11.
$$

You have already seen evaluation of algebraic expressions in earlier chapters; here the focus is that $f$ is specifically a polynomial function, so the output is always a finite sum of powers of the input.

Shape and Smoothness (Qualitative Overview)

Without going into detailed graphing (which comes later), there are two important, general geometric features of polynomial functions:

Domain: For a polynomial function $f(x)$, you can plug in any real number $x$. There are no restrictions like “$x \neq 0$” or “$x > 0$” that often appear with rational or root functions. So the domain is all real numbers.
Smoothness and continuity: The graph of a polynomial function is a smooth, unbroken curve:

no jumps,
no holes,
no sharp corners or cusps.

These properties make polynomial functions easier to work with and are important in later topics, such as calculus.

Summary of Key Points

A polynomial function in one variable is a function of the form
$$
f(x) = a_n x^n + \dots + a_1 x + a_0
$$
with real coefficients and nonnegative integer exponents.
Terms like $x^{-1}$, $\sqrt{x}$, $2^x$, $\sin x$, or expressions with $x$ in denominators or inside roots/logs are not polynomial.
Constant functions (like $f(x) = 7$) and the zero function ($f(x) = 0$) are special cases of polynomial functions.
Writing a polynomial in standard form (descending powers of $x$) and combining like terms makes its structure clearer.
Polynomial functions are defined for all real $x$ and have graphs that are continuous and smooth.

Later sections on degree and behavior and zeros of functions will build on this foundation to describe how polynomial functions behave for large $|x|$ and how they intersect the $x$-axis.