Lesson Explainer: Polynomial Functions

In this explainer, we will learn how to identify, write, and evaluate a one-variable polynomial function and state its degree and leading coefficient.

Polynomial functions appear all throughout science and in many real-world applications including mechanics and finance. For example, the area and volume of many shapes can be calculated by using polynomial functions. If we consider the function , this can be used to calculate the area of a square with a side length . Here, the output of the function would be equal to the area of a square with side length .

To fully understand what is meant by a polynomial, we need to start with the building blocks of polynomials; these are known as monomials.

Definition: Monomials

A monomial is a product of constants and variables, where the variables may contain only nonnegative integer exponents.

For example, is a monomial since it is a single term in which, every variable has only nonnegative integer exponents. To better understand exactly what is meant by a monomial, let’s have a look at a list of expressions and determine which are monomials:

We can rewrite option A to be , which is a positive integer, so this is a monomial.

Option B contains the variable raised to the power of three. It does not matter what we name our variables, so this is also a monomial.

In option C, we recall that we can rewrite a square root as a power of ; this allows us to write option C as . Since this is not an integer, we can say that this expression is not a monomial.

In option D, we note that 0 is an example of a monomial since we can write it as . Similarly, 1 is an example of a monomial since it can be written as . In fact, any constant value is a monomial, since it can be written as .

In option E, we can deduce that is not a monomial since the expression contains multiple terms; it is, however, a sum of monomials.

In option F, we see that contains a negative exponent, so it is not a monomial.

Finally, in option G, we note that is a monomial since it is a single term and every variable is raised to an integer exponent. The fact that the constant is not an integer does not matter as we only require the powers of the variables to be integers.

We are now ready to define polynomials using our understanding of monomials.

Definition: Polynomial Functions

A polynomial is an expression that is a sum of monomials, where each term is called a monomial term. A function that is polynomial is called a polynomial function.

For example, we saw that was not a monomial, but it is a polynomial since it is the sum of two monomials. We can also say that is a polynomial function, where it is worth noting that we often leave out the word function and just say is a polynomial. We also call this a one-variable polynomial since there is only one variable that appears in the polynomial. For the rest of this explainer, we will focus entirely on one-variable polynomials; each term is a product of constants and a single variable, where the variable must have a nonnegative integer exponent.

It is worth noting that any function in the form for a constant value is an example of one-variable polynomials since we can write these functions in terms of a single variable. For example, can be written as

To help us understand the concept of polynomials, let’s determine which of the following expressions are polynomials:

In options A and B, every single term is a monomial, so these are both polynomials. It is worth noting that all monomials are polynomials containing a single term. In option C, we know that , so this is also a polynomial.

In option D, we can write the term as . Since the exponent is negative, this term is not a monomial, and hence the expression is not a polynomial. Finally, in option E, is a one-variable polynomial in the variable .

Let’s now see an example of determining which of a list of functions is a polynomial function.

Example 1: Identifying a Polynomial Function

Which of the following is a polynomial function?

Answer

We recall that a single-variable polynomial function is one where every term is a product of constants and a single variable, where the variable must have nonnegative integer exponents.

Let’s go through each option individually. First, we see that option A contains the term . By using the laws of exponents, we can write this as . This is a variable raised to a noninteger exponent, so this is not a polynomial function.

Second, we see that options B and D contain the expression . This is a variable raised to a negative exponent, so these are not polynomial functions.

Third, option C is the function . Using the laws of exponents, we can rewrite this as . This is a variable raised to a negative exponent, so this is not a polynomial function.

Finally, each term in option E is a product of constants and a single variable, where the variable must have nonnegative integer exponents. For example, we can write , so option E is a polynomial.

Only option E, , is a polynomial function.

In our next example, we will construct a polynomial function from given information in a real-world problem.

Example 2: Writing a Polynomial Function

A bus service charges a fixed fee of 5 pounds and an additional 2 pounds for every bus stop passed. Write a polynomial function to represent the cost of a ride.

Answer

To construct a function that represents the cost of a ride, we first need to determine exactly how the cost of the ride is calculated. We are told there is a fixed fee of 5 pounds and an additional cost of 2 pounds for every bus stop passed, so let’s call the number of every bus stop passed .

Then, if we pass bus stops, we would need to pay plus the fixed fee of 5 pounds, for a total cost of . Writing this in function notation, we have where, it is worth noting, the value of must be a positive integer, since it represents the number of bus stops passed. This means the domain of is the set of positive integers.

It is worth checking that this is a polynomial function, since the question explicitly asks for a polynomial function. To do this, we first recall that a single-variable polynomial function is one where every term is a product of constants and a single variable, where the variable must have nonnegative integer exponents. We can use the law of exponents to rewrite the function as where we see that the only exponent of a variable is 1, which is a nonnegative integer. Hence, this is a polynomial function, and the cost of the bus journey where is the number of bus stops passed is given by

In our next example, we will evaluate a polynomial function at a given value. To do this, we recall that we can evaluate any function at a value by using substitution. For a function , we evaluate by substituting into the function .

Example 3: Evaluating a Polynomial Function

If , find .

Answer

We are asked to determine the value of . We recall that this is function notation for the value of when . Hence, we can find this value by substituting into the function; this gives

Therefore, .

Before we move on to more examples involving polynomial functions, we can discuss some useful terminology to help us describe the type of polynomial function we are working with.

Definition: Degree, Leading Term, and Leading Coefficient of a Single-Variable Polynomial

For a single-variable polynomial, we define the following:

The largest exponent of a variable in any nonzero term is called its degree.
The term with highest degree in a polynomial is called its leading term.
The constant factor of the leading term is called the leading coefficient.

For example, consider the polynomial function

Since this is a single-variable polynomial, its degree is the largest exponent in any nonzero term. We can use the laws of exponents to write each term in terms of the exponents of :

First, we note that , so this term is not included in the calculation of the degree. We rewrite the function as

Next, the largest of these exponents is 3, so we say that this polynomial has degree 3. The term containing is then the leading term; this is . Finally, the constant factor of this term, 5, is the leading coefficient.

There are a few types of polynomials where it can be difficult to determine these values. For example, consider the polynomial function

We can write this function in terms of the variable as

Then, we have that the largest exponent of in a nonzero term is 0, so the degree of the polynomial is 0. The leading term is the only term, , and the leading coefficient is also .

Finally, we have to make a special case about the zero polynomial, which, as a one-variable polynomial function, can be written as

The degree of this polynomial would be the largest exponent of a variable in any nonzero term. However, every term in this function is zero, so we leave the degree of this polynomial undefined.

The degree of a polynomial tells us information about its shape and complexity, and as such, we give names to the families of polynomials based on their degree.

Key Terms: Names of Polynomial Functions Based on Their Degree

For one-variable polynomial functions, we have the following:

A degree 0 polynomial function is called a constant function.
A degree 1 polynomial function is called a linear function.
A degree 2 polynomial function is called a quadratic function.
A degree 3 polynomial function is called a cubic function.
A degree 4 polynomial function is called a quartic function.
A degree 5 polynomial function is called a quintic function.

Degree 0 polynomial functions are called constant functions because they do not vary as the value of the input varies; in a single-variable polynomial, they are all of the form for some constant . It is also worth noting that there are names for polynomials of higher degrees, but beyond degree 5, these names are not commonly used.

Let’s now see an example of using these definitions to determine the degree and leading coefficient of a given polynomial function.

Example 4: Identifying the Degree and Leading Coefficient of a Polynomial Function

Find the degree and leading coefficient of the polynomial function .

Answer

We recall that, in a single-variable polynomial, the largest exponent of a variable in any nonzero term is called its degree, the term with highest degree in a polynomial is called its leading term, and the constant factor of the leading term is called the leading coefficient.

Since the only variable in this polynomial is , we need to determine the largest exponent of that appears in a nonzero term. By using the laws of exponents, we can rewrite the function as follows:

We can see that the largest exponent of is 4, so the degree of this polynomial function is 4. The term containing is then called the leading term; in this case, this is the term . Finally, the constant factor in the leading term is the leading coefficient and the constant factor in is 3.

Hence, the degree of is 4 and the leading coefficient is 3.

In our next example, we will determine the type of polynomial function we are given.

Example 5: Identifying the Type of Polynomial Function

Identify the name of the polynomial function .

Answer

We recall that we name polynomial functions based on their degree and that, in a single-variable polynomial, the largest exponent of a variable in any nonzero term is called its degree. We can determine the degree of this polynomial by using the laws of exponents to rewrite the function as follows:

The largest exponent of is 3, so this is the degree of the polynomial. Finally, we recall that we call all single-variable degree 3 polynomial functions cubic functions.

Hence, is a cubic function.

In our final example, we will evaluate a polynomial function at an algebraic expression.

Example 6: Substituting a Quadratic Expression into a Cubic Function

Consider the polynomial function . Evaluate .

Answer

Here, we need to substitute a quadratic expression into a cubic function. To do this, we need to replace the variable , in the cubic function, with the expression . This gives us

We could leave our answer like this; however, it is good practice to simplify any function where possible. We will distribute the exponents over the parentheses to get

Finally, we will distribute the coefficients over the parentheses and collect like terms to get

Hence, .

Let’s finish by recapping some of the important points of this explainer.

Key Points

A monomial is a product of constants and variables, where the variables may only contain nonnegative integer exponents.
A polynomial is an expression that is the sum of monomials, where each term is called a monomial term. Every term will be the product of constants and variables, where the variables may only contain nonnegative integer exponents.
A single-variable polynomial is a polynomial that contains only a single variable.
The largest exponent in any nonzero term in a single-variable polynomial is called its degree, the term with the highest degree in a polynomial is called its leading term, and the constant factor of the leading term is called the leading coefficient.
Polynomial functions have different names based on their degree. For one-variable polynomial functions, we have the following:
- A degree 0 polynomial function is called a constant function.
- A degree 1 polynomial function is called a linear function.
- A degree 2 polynomial function is called a quadratic function.
- A degree 3 polynomial function is called a cubic function.
- A degree 4 polynomial function is called a quartic function.
- A degree 5 polynomial function is called a quintic function.