Lesson Video: Monomials Mathematics

In this video, we will learn how to determine whether an expression is a monomial.

09:20

Video Transcript

In this video, we will learn how to determine whether an expression is a monomial. We will learn how to tell the difference between monomials, binomials, and trinomials. And we will also find the degree of a monomial. We will begin with some key definitions. A monomial is a polynomial which has only one term. It can be a number, a variable, or a product of numbers and variables. All exponents or powers in a monomial must be nonnegative integers. Examples of monomials are 42, five 𝑥, three 𝑦 to the power of four, and two 𝑝 squared 𝑞.

The degree of a monomial is the sum of the exponents of all included variables. Constants have the monomial degree of zero. This means that the number 42 has a degree of zero. As five 𝑥 is the same as five 𝑥 to the power of one, this has a degree of one. Three 𝑦 to the fourth power has a degree of four. In the monomial two 𝑝 squared 𝑞, the 𝑞 has a power or exponent of one. As two plus one is equal to three, the degree of two 𝑝 squared 𝑞 is three. As the word poly means many or several, we usually think of a polynomial as having more than one monomial. A polynomial is therefore the sum of monomials, where each monomial is known as a term. We will now briefly look at some different types of polynomials.

A binomial has two terms. This means it is made up of two monomials. Examples of binomials are 𝑥 plus four and seven 𝑥 squared minus nine. A trinomial has three terms. This is made up of three monomials, for example, 𝑥 squared plus seven 𝑥 minus nine and four 𝑥 to the fifth power plus seven 𝑥 plus one. The degree of a polynomial is the greatest degree of its terms. In these four examples, the degree of the polynomial will be one, two, two, and five, respectively. We will now look at some questions involving monomials.

James claims that any number, for example, 213, is a monomial. Is James correct?

We know that the definition of a monomial states that it could be a number, a variable, or product of numbers and variables. This means that any number or constant is a monomial. The correct answer is yes, James is correct.

Our next question involves solving a problem about the properties of a monomial.

Given that the expression 𝑥 to the power of 𝑎 𝑦 to the power of 𝑏 is a monomial. What must be true of the exponents 𝑎 and 𝑏? Is it option (A) 𝑎 and 𝑏 are both negative integers or option (B) 𝑎 and 𝑏 are both nonnegative integers?

We recall that a monomial is a number, a variable, or, as in this case, a product of numbers and variables. We have two variables, 𝑥 to the power of 𝑎 and 𝑦 to the power of 𝑏, being multiplied together. There is, however, a second part to our definition. This states that all exponents must be nonnegative. The correct answer is option (B) 𝑎 and 𝑏 are both nonnegative integers. This means that 𝑥 to the fourth power 𝑦 cubed and 𝑥 to the ninth power 𝑦 to the 14th power are both monomials. If one or both of the exponents were negative, the expression would not be a monomial, for example, 𝑥 to the power of negative two 𝑦 cubed. An expression where one of the exponents was a fraction, such as one-third, would also not be a monomial.

The next question looks at identifying the properties of a monomial.

Chloe says that a monomial is an expression that contains a single variable, for example, 𝑥 plus 𝑥 squared. Is Chloe correct?

A monomial is a number, a variable, or the product of numbers and variables. This means that it must be a single term. The example given after Chloe’s statement, 𝑥 plus 𝑥 squared, has two terms. This means it is a binomial. Whilst it is possible that a monomial could have a single variable, the correct definition is that a monomial is an expression that contains a single term. This means that her statement is incorrect, and the correct answer is no.

Examples of monomials are the constant or number seven, eight 𝑥, and five 𝑥 squared. Eight 𝑥 and five 𝑥 squared do have a single variable. However, other examples of monomials are four 𝑥𝑦 and 𝑥 cubed 𝑦 squared, which both have two variables. It is not the number of variables that dictates whether an expression is a monomial, but rather the number of terms.

Our last two questions will involve looking at the degree of a monomial.

Find the degree of the monomial negative seven.

The degree of any monomial is the sum of the exponents or powers of all included variables. This might lead us into thinking that our degree is one as negative seven to the power of one is equal to negative seven. However, negative seven is not a variable; it is a constant. This leads us to the second part of our definition of the degree of monomials, which states that all constants have the monomial degree of zero. The degree of negative seven is therefore zero.

Find the value of 𝑛 if the degree of the monomial negative 𝑥 to the power of 𝑛 𝑦 is six.

We recall the definition that the degree of a monomial is the sum of the exponents of all included variables. We also note that constants have the monomial degree of zero. The variable 𝑦 is the same as 𝑦 to the power of one. This means that the variable 𝑥 has an exponent of 𝑛, and the variable 𝑦 has an exponent of one. We are told in the question that the degree of the monomial is six. Therefore, 𝑛 plus one is equal to six. Subtracting one from both sides of this equation gives us 𝑛 is equal to five. The value of 𝑛 if the degree of the monomial negative 𝑥 to the power of 𝑛 𝑦 is six is 𝑛 is equal to five.

We will now summarize the key points from this video. We found that a monomial is a special type of polynomial that has only one term. They can be numbers, variables, or products of numbers and variables. We also found out that the degree of a monomial is the sum of the exponents of all the variables. Constants have the monomial degree zero.

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