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Lesson Explainer: Monomials, Binomials, and Trinomials Mathematics • 9th Grade

In this explainer, we will learn how to classify polynomial expressions as monomials, binomials, or trinomials and how to find the degrees of these expressions.

Polynomial expressions appear throughout mathematics, statistics, and physics. They are used in a number of applications, including the area of a shape or motion. Before we discuss polynomials, we should start with the definition of the building blocks of polynomials, known as 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.

The simplest example of a monomial is a constant term. For example, 2 is a monomial. We can also multiply this constant by variables with nonnegative integer exponents to construct other monomials. To fully understand the concept of a monomial, let’s consider the following list of expressions and determine which are monomials:

  1. 3π‘¦οŠ¨
  2. π‘₯
  3. π‘₯+π‘₯
  4. 32π‘₯π‘¦οŠ¨
  5. π‘₯

The expression labeled A is the product of the constant 3 and the variable 𝑦 raised to an exponent of 2, which is a nonnegative integer, so this is a monomial. However, expression B is not a monomial since the exponent βˆ’1 is a negative integer. Expression C is also not a monomial since it contains two terms. Expression D is a monomial; we can see that it is a single term that is a product of a constant and two variables each raised to a nonnegative integer exponent. Finally, expression E is not a monomial since the exponent 12 is not an integer.

Let’s now see an example of determining which of a list of expressions are monomials.

Example 1: Identifying Whether an Expression is a Monomial

Which of the following expressions are monomials?

  1. π‘₯
  2. π‘₯π‘¦οŠ©οŠͺ
  3. π‘₯
  4. 1𝑦
  5. 1+π‘₯

Answer

We start by recalling that a monomial is a single-term algebraic expression that is the product of constants and variables, where the variables may only have nonnegative integer exponents.

We see that, in expression A, the only variable, π‘₯, has an exponent of 2, which is a positive integer, so the expression is a monomial. Similarly, in expression B, both variables have nonnegative integer exponents and the expression is a single term, so it too is a monomial.

In expression C, we note that the exponent is negative, so the expression is not a monomial. In expression D, we can rewrite 1𝑦 as π‘¦οŠ±οŠ§, which has a negative exponent, so the expression is not a monomial. Finally, we can see that expression E has two terms, so the expression is not a monomial.

Hence, of the given expressions, only expressions A and B are monomials.

Before we move on, we can describe the different parts of a monomial by using the following definitions.

Definition: Degree and Coefficient of a Monomial

  • The degree of a monomial is the sum of the exponents of its variables.
  • The constant factor in a monomial is called its coefficient.

For example, in the monomial 53π‘₯π‘¦οŠ¨, we can see the constant factor is 53; this is called its coefficient. We can also find the degree of the monomial by adding the exponents of the variables. We note that the exponent of π‘₯ is 1 and the exponent of 𝑦 is 2, so the degree of this monomial is 2+1=3.

Let’s now see an example of determining the degree of a constant monomial.

Example 2: Finding the Degree of a Constant Monomial

Find the degree of the monomial βˆ’7.

Answer

We recall that the degree of a monomial is the sum of the exponents of the variables. We see that the monomial βˆ’7 has no variables. So, to find the sum of these exponents, we first note that βˆ’7=βˆ’7π‘₯ or indeed βˆ’7=βˆ’7π‘₯π‘¦οŠ¦οŠ¦ or any number of variables raised to exponents of 0.

Since the exponents of the variables are 0, the sum of these exponents is also 0. Hence, the degree of the monomial is 0.

It is worth noting that the working in the above example is the same for any constant monomial. We can show that if 𝑐 is a constant, then the degree of 𝑐 is 0.

Let’s now see one more example of determining the degree of a monomial.

Example 3: Finding the Degree of a Monomial

  1. Find the degree of the monomial βˆ’3π‘₯.
  2. Find the degree of the monomial 7π‘₯π‘¦π‘§οŠ¨.

Answer

We first recall that the degree of a monomial is the sum of the exponents of the variables.

Part 1

Since βˆ’3π‘₯ only has a single variable π‘₯, its degree is just the exponent of π‘₯, which in this case is 5.

Part 2

In this case, we have 3 variables, π‘₯, 𝑦, and 𝑧, and we can rewrite 7π‘₯π‘¦π‘§οŠ¨ as 7π‘₯π‘¦π‘§οŠ§οŠ§οŠ¨. Hence, we can see that the exponents of the variables are 1, 1, and 2 respectively. Summing these exponents gives 1+1+2=4.

Therefore, the degree of the monomial 7π‘₯π‘¦π‘§οŠ¨ is equal to 4.

We are now ready to define polynomials and a few specific types of polynomials.

Definition: Binomials, Trinomials, and Polynomials

The sum of two monomial terms that cannot be simplified into a single term is called a binomial.

The sum of three monomial terms that cannot be simplified into an expression with two or fewer terms is called a trinomial.

The sum of any number of monomial terms is a polynomial.

Since a polynomial is the sum of any number of monomials, we can first note that monomials, binomials, and trinomials are all examples of polynomials.

Example 4: Identifying a Trinomial from a List of Different Polynomials

Consider this list of expressions:

  1. βˆ’1βˆ’π‘₯+1
  2. π‘₯+𝑦
  3. 2π‘₯π‘¦βˆ’3π‘₯+1οŠͺ
  4. π‘₯+π‘₯+2π‘₯
  5. βˆ’3π‘₯+2π‘₯+0

Which expression (or expressions) is a trinomial?

Answer

We first recall that a trinomial is the sum of three monomial terms that cannot be simplified into two or fewer terms and also that a monomial is a single-term algebraic expression that is the product of constants and variables, where the variables only have nonnegative integer exponents.

We can check each expression separately to determine whether it is a trinomial.

We start with expression A. Although each term in this expression is a monomial and there are three terms, we note that we can simplify the expression since βˆ’1+1=0. So, βˆ’1βˆ’π‘₯+1=βˆ’π‘₯.

Since this does not have three terms, it is not a trinomial.

In expression B, we see there are only two terms; since these are both monomials, we can say that this is a binomial but not a trinomial.

In expression C, we have three terms and each is the product of constants and variables raised to nonnegative integer exponents. Hence, expression C is a trinomial.

In expression D, we note that the exponent of π‘₯ in the second term is not an integer. So, this term is not a monomial. Therefore, this expression is not the sum of monomials, so it cannot be a trinomial.

In expression E, we note that this is the sum of three monomial terms. However, adding 0 will not change the value, so we can write this as βˆ’3π‘₯+2π‘₯, which is a binomial. Hence, this expression is not a trinomial.

Hence, only expression C is a trinomial.

In our next example, we will determine the value of an unknown from the given degree of a monomial.

Example 5: Finding the Missing Exponent given the Degree of the Monomial

Find the value of 𝑛 if the degree of the monomial (βˆ’π‘₯𝑦) is 6.

Answer

We recall that the degree of a monomial is the sum of the exponents of the variables. Since 𝑦 can be written as π‘¦οŠ§, the exponent of π‘₯ is 𝑛 and the exponent of 𝑦 is 1. So, the degree of the monomial is the sum of these values: 𝑛+1.

Since we are told that the degree of the monomial is 6, we must have 6=𝑛+1.

Subtracting 1 from both sides of the equation yields 𝑛=5.

We can also introduce the following definitions to help us describe different polynomial expressions.

Definition: Degree of a Polynomial

  • The degree of a polynomial (including binomials and trinomials) 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.

In our final example, we will apply these definitions to determine which of 5 given expressions are binomials of degree 3.

Example 6: Identifying a Binomial with a Specific Degree from a List of Polynomials

Consider this list of expressions:

  1. π‘₯𝑦𝑧+π‘₯
  2. π‘₯+3π‘₯βˆ’2π‘₯
  3. π‘₯𝑦+3
  4. π‘¦οŠ©
  5. π‘Žπ‘+π‘οŠ¨οŠ©

Which expression (or expressions) is a binomial of degree 3?

  1. d
  2. c
  3. b, c, and e
  4. e and b
  5. a and e

Answer

We begin by recalling that a binomial is the sum of two monomial terms that cannot be simplified into a single term and the degree of a binomial is the greatest sum of the exponents of the variables in a single term.

We can check each expression separately to determine whether it is a binomial and, if so, whether its degree is 3.

In expression a, we note that there are two terms, each of which is the product of variables raised to nonnegative integer exponents. So, this expression is the sum of two monomials; hence, it is a binomial. To calculate the degree of this binomial, we note that the term π‘₯ has degree 2, and the term π‘₯𝑦𝑧 can be written as π‘₯π‘¦π‘§οŠ§οŠ§οŠ§ and so its degree is 1+1+1=3. The largest of these degrees is 3, so the degree of this expression is 3. Hence, this is a binomial of degree 3.

In expression b, we note that there are three monomial terms that cannot be simplified into fewer terms. Thus, this expression is a trinomial and not a binomial.

In expression c, we see that there are two monomial terms, so this is a binomial. However, π‘₯𝑦 has degree 2 and the constant term, 3, has degree 0. So, the term with the highest degree in this expression has degree 2, which means the degree of this expression is 2. Therefore, this expression does not have degree 3.

In expression d, we see that there is only a single monomial term, so this expression is not a binomial.

In expression e, we see that there are two monomial terms and the degree of both terms is 3, so this is a binomial of degree 3.

Hence, only expressions a and e are binomials of degree 3.

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 only have nonnegative integer exponents.
  • The sum of two monomial terms that cannot be simplified into a single term is called a binomial.
  • The sum of three monomial terms that cannot be simplified into two or fewer terms is called a trinomial.
  • The sum of any number of monomial terms is a polynomial.
  • The degree of a monomial term is the sum of the exponents of the variables.
  • The constant factor in a monomial is called its coefficient.
  • The degree of a polynomial (including binomials and trinomials) is the greatest degree of any of its monomial terms.

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