In this explainer, we will learn how to determine the degree of a polynomial and use the terminology associated with polynomials, such as terms, coefficients, and constants.

Polynomials appear throughout mathematics, having uses in optimization problems, projectile mechanics, and finance, to name a few. To understand what is meant by polynomials, we first have to describe the building blocks of polynomials, which are called monomials.

### Definition: Monomial

A monomial is a single-term algebraic expression that is the product of constants and variables, where the variables only have nonnegative integer exponents.

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

- 0

We can see that the expression in option a is a single term, and we can write this expression as . Since the variable is raised to a nonnegative integer exponent, this expression is a monomial.

In expression b, we note that there are two nonzero terms. Hence, this is not a monomial but the sum of two monomial expressions.

Expression c can be rewritten as and since is not an integer, this expression is not a monomial.

Expression d can be rewritten as and since the exponent is negative, this is not a monomial.

For expression e, we note that 0 can be written as , so 0 is a monomial. Similarly, 1 is an example of a monomial since it can be written as . In fact, any constant is a monomial since it can be written as .

Finally, we can see that expression f is a single term and that each variable is raised to a nonnegative integer exponent. Hence, this expression is a monomial. It is worth noting that the constant factors can have any exponents since we only restrict the exponents for the variables. This is why we can have a factor of in the monomial in expression f.

Before we move on to define polynomials, we introduce one piece of terminology for monomials. We refer to the constant factor of a monomial as its coefficient. For example, has a coefficient of 2, and has a coefficient of .

We are now ready to define polynomials using monomials.

### Definition: Polynomials

A polynomial is an expression that is the sum of monomials, where each monomial is called a monomial term. The number of monomials in the expression is called the number of terms of the polynomial.

For example, we saw that is the sum of two monomial terms, which means it is also a polynomial. We call this a one-variable (or single-variable) polynomial since it only contains one variable, .

It is also worth noting that all monomials are polynomials since we only require each term in our polynomial to be a monomial. In particular, this means we have shown that all constants are polynomials.

Before we move on to discuss terminology, let’s determine which expressions in the following list are polynomials to help us better understand this concept:

In expression a, we can see that the second term can be written as . Since this is a variable raised to a noninteger exponent, this expression is not a polynomial.

In expression b, each term is the product of constants and variables raised to nonnegative integer exponents, so each term is a monomial. Finally, we can consider the difference of these terms as the sum: . So, this is a polynomial.

Finally, in expression c, we can write ; since variable is raised to a negative exponent, this is not a polynomial.

Before we move on to example questions involving polynomials, we can discuss some further useful terminology to help us describe the type of polynomial we are working with.

### Definition: Degree, Leading Term, and Leading Coefficient of a Polynomial and Monomial

- The degree of a monomial term is the sum of the exponents of the variables.
- The degree of a polynomial is the greatest degree of any of its monomial terms. Equivalently, we can say that the degree of a polynomial is the greatest sum of the exponents of the variables in any single term of the polynomial.
- The term with the highest degree in a polynomial is called its leading term.
- The coefficient of the leading term is called the leading coefficient.

Let’s use these definitions to determine the degree, leading term, and leading coefficient of the polynomial .

Firstly, to determine the degree, we need to find the sums of the exponents of the variables in the nonzero terms. The exponent of in the first term is 2, and . So, the exponent of is 1. This means the sum of the exponents of the variables in the first term is . Hence, the degree of the first monomial term is 3.

We apply the same process to the second term. We see that the exponents of the variables are 1, 2, and 1, so the degree of the second term is 4. The monomial term with the largest degree is 4, so the polynomial has a degree of 4.

Secondly, we see that the term with the highest degree is , so this is the leading term.

Thirdly, we can see that this polynomial contains two monomial terms; we can say that this is a polynomial with 2 terms.

Finally, the coefficient of the leading term is the constant factor, which is in this case. Hence, the leading coefficient of this polynomial is .

Let’s now see an example of how to find the degree of a single-variable polynomial.

### Example 1: Finding the Degree of a Polynomial

Determine the degree of .

### Answer

We recall that the degree of a polynomial is the greatest sum of the exponents of the variables in any single term of the polynomial. Since the given polynomial only contains a single variable, this sum will only include a single exponent. Therefore, we only need to look for the largest exponent of the variable . We see that this is in the term , which has an exponent of 4. Hence, we can say that the polynomial has a degree of 4 and that this is a polynomial of the fourth degree.

In our previous example, we saw a useful property: in a single-variable polynomial, the sum of the exponents of the variable in each term will just be the exponent of the only variable. This then allowed us to conclude that the degree of any single-variable polynomial is the greatest exponent of the variable in a nonzero term.

In our next few examples, we will see a few other useful descriptions of parts of a polynomial expression.

### Example 2: Identifying the Constant Term in a Polynomial

What is the constant term in the expression ?

### Answer

The constant term in any expression is the term that remains constant. In other words, it contains no variables. We can see that the first term, , contains the variable and that the second term, , contains the variable , so these are not constant terms. The third term is 27, which contains no variables.

Hence, the constant term in the given expression is 27.

Let’s now practice identifying the coefficient of a term in a single-variable polynomial.

### Example 3: Finding the Coefficient of a Term in a Polynomial

What is the coefficient of in the expression ?

### Answer

We recall that we refer to the constant factor of a monomial as its coefficient. Therefore, the question is asking for the constant factor of in the expression . We can answer this by noting that , so it has a constant factor of 1.

Hence, the coefficient of in the given expression is 1.

In our next example, we will determine the coefficient and degree of a monomial.

### Example 4: Finding the Degree and Coefficient of a Polynomial Term

Determine the coefficient and the degree of .

### Answer

We start by noting that this is a single term that is the product of constants and variables raised to nonnegative integer exponents, so this is a monomial term. We recall that the coefficient of a single term is its constant factor. Since is a variable, the coefficient is .

We also recall that the degree of a monomial is the sum of the exponents of the variables. In this case, there is a single variable with an exponent of 3, so this sum only consists of the exponent 3. Hence, it has a degree of 3.

Therefore, the coefficient is , and the degree is 3.

In our final example, we will determine which expression in a given list has the same degree as a given polynomial.

### Example 5: Identifying Polynomials of the Same Degree

Which of the expressions below is of the same degree as ?

### Answer

We start by noticing that the given expression and the expressions in the choices are the sum of the products of constants and variables raised to nonnegative integer exponents. In other words, they are all polynomials. We can then recall that the degree of a polynomial is the greatest sum of the exponents of the variables in any single term of the polynomial.

Hence, we can answer this question by determining the degree of all five polynomials. Let’s start with the degree of the polynomial given in the question. We do this term by term by adding the exponents of the variables. The first term contains only a single variable raised to an exponent of 8, so the degree of this term is 8. The second term has two variables with exponents of 4 and 2. We add these to see that the degree of this term is . Finally, the third term contains only a single variable, so its degree is the value of its exponent, which is 2. The largest of these degrees is 8, so we need to determine which of the four choices is an eighth-degree polynomial.

Let’s now determine the degree of each of the choices separately.

In choice A, , the first term is a single variable, so it has a degree equal to the value of the exponent of , which is 4. The second term has a degree of . We could then find the degree of the third term; however, it is not necessary since we have shown that the degree of this polynomial is at least 11.

In choice B, , the first term has a degree of 7, the second term has a degree of , and the third term has a degree of 2, so this is a polynomial of degree 7.

In choice C, , we can notice that the first term has a degree of 9, so this polynomial will have a degree of at least 9, which is greater than 8. So, this is not the same degree as that of the polynomial given.

Finally, in choice D, , the first term has a degree of 2, the second term has a degree of , and the third term has a degree of 7. The greatest of these is 8, so this is also a polynomial of degree 8.

Hence, the answer is choice D.

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

### Key Points

- A monomial is a single-term algebraic expression that is the product of constants and variables, where the variables can have nonnegative integer exponents.
- A polynomial is an expression that is the sum of monomials, where each term is called a monomial term.
- A single-variable polynomial is a polynomial that contains only a single variable.
- The constant factor of a monomial term is called its coefficient.
- The degree of a polynomial is the greatest sum of the exponents of the variables in any single term of the polynomial.
- The term with the highest degree in a polynomial is called its leading term.
- The coefficient of the leading term is called the leading coefficient.