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Polynomial Functions (College Board AP® Precalculus): Study Guide

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 9 April 2026

Polynomial functions

What is a polynomial function?

A polynomial function of $x$ is any function that can be written in the form $p (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + a_{n - 2} x^{n - 2} + \dots + a_{2} x^{2} + a_{1} x + a_{0}$
- where $n$ is a positive integer
- and $a_{i}$ is a real number for each $i$ from 0 to $n$ , with $a_{n} \neq 0$

Examiner Tips and Tricks

You don't need to memorize this formal definition, but you do need to be able to recognize polynomial functions and understand the key terminology.

What are the key terms for polynomial functions?

The degree of a polynomial is the highest power of $x$ that appears in the expression
- E.g. $p (x) = 3 x^{4} - 2 x^{2} + x - 7$ has degree $4$
The leading term is the term with the highest power of $x$
- E.g. for $p (x) = 3 x^{4} - 2 x^{2} + x - 7$ , the leading term is $3 x^{4}$
The leading coefficient is the coefficient of the leading term
- E.g. for $p (x) = 3 x^{4} - 2 x^{2} + x - 7$ , the leading coefficient is $3$
These three properties (degree, leading term, and leading coefficient) are important
- They determine much of the polynomial's behavior
  - including its end behavior and general shape

How do familiar function types relate to polynomials?

Many function types that you already know are special cases of polynomial functions:
- A constant function, such as $f (x) = 5$ , is a polynomial of degree 0
  - It can be thought of as $f (x) = 5 x^{0} = 5$
- A linear function, such as $f (x) = 2 x + 3$ , is a polynomial of degree 1
- A quadratic function, such as $f (x) = - x^{2} + 4 x - 1$ , is a polynomial of degree 2

Polynomials of higher degree include:
- Cubic functions (degree 3), e.g. $p (x) = 2 x^{3} - x + 4$
- Quartic functions (degree 4), e.g. $p (x) = x^{4} - 3 x^{3} + 2 x$
There is no limit to the maximum degree a polynomial function can have

How can I recognize a polynomial function?

A polynomial function must be a sum (or difference) of terms
- where each term consists of a constant multiplied by a non-negative integer power of $x$
- The constant may be 1
  - e.g. $x^{3} = 1 x^{3}$
The following are not polynomial functions:
- $f (x) = \frac{1}{x}$
  - this involves a negative power of $x$ (i.e. $x^{- 1}$ )
- $f (x) = \sqrt{x}$
  - this involves a fractional power of $x$ (i.e. $x^{1 / 2}$ )
- $f (x) = 2^{x}$
  - this is an exponential function, not a polynomial
- $f (x) = \sin (x)$
  - this is a trigonometric function, not a polynomial

Does the order of terms matter?

A polynomial does not have to be written with the terms in order of decreasing degree (or in any particular order)
- E.g. $p (x) = 1 - 3 x + 5 x^{3}$ is the same polynomial as $p (x) = 5 x^{3} - 3 x + 1$
- The degree is still $3$ , the leading term is still $5 x^{3}$ , and the leading coefficient is still $5$
To identify the degree and leading term, look for the term with the highest power of $x$ , regardless of where it appears in the expression

Worked Example

For each of the following polynomial functions, identify the degree, the leading term, and the leading coefficient.

(i) $f (x) = 7 x^{3} - 4 x^{5} + 2 x - 9$

(ii) $g (x) = 6 - x + 3 x^{2}$

(iii) $h (x) = 11$

Answer:

(i)

The term with the highest power of $x$ is $- 4 x^{5}$

Degree: $5$
Leading term: $- 4 x^{5}$
Leading coefficient: $- 4$

(ii)

The term with the highest power of $x$ is $3 x^{2}$

Degree: $2$
Leading term: $3 x^{2}$
Leading coefficient: $3$

(iii)

The function $h (x) = 11$ is a constant function

It may be written equivalently as $11 x^{0}$
A constant is a polynomial of degree $0$

Degree: $0$
Leading term: $11$
Leading coefficient: $11$

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