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.
