Video Transcript
Degree and Coefficient of
Polynomials
In this video, we’ll learn what we
mean by a polynomial and we’ll define several different words to help us describe
different parts of polynomials. We’ll learn what we mean by the
degree of a polynomial, what we mean by the coefficients of different parts of a
polynomial, and we’ll see how we can find these given a polynomial.
To do this, we’re going to start by
defining the building blocks of polynomials. These are called monomials. And a monomial is an expression
which consists just of a product of constants and variables, where it’s important to
know our variables can only have nonnegative integer exponents. We can then give an example of some
monomials. For example, two 𝑥 is a monomial
because it’s a product between the constant two and 𝑥. And remember, 𝑥 is just 𝑥 to the
first power. Another example might be negative
𝑦 squared. 𝑦 is a variable, so we’re allowed
to raise this to the power of two. And remember, negative 𝑦 squared
is negative one times 𝑦 squared. So, this is another example of a
monomial.
Another example is any
constant. For example, we could just take the
constant three. And it’s important to realize we’re
allowed to take any exponent of our constants. For example, we could take the
square root of three. This is also an example of a
monomial. One last example of a monomial is
the square root of two times 𝑥 times 𝑦 squared. It’s important to realize we’re
allowed to have multiple variables in our monomials as long as our exponents are
nonnegative integers. Now that we’ve defined the
monomial, we’re ready to define a polynomial.
A polynomial is just an expression
which is the sum of one or more monomials. In other words, we create
polynomials by adding together multiple monomials. So, to construct some examples of
polynomials, we can use our monomials. The first thing we can notice is a
polynomial is the sum of one or more monomial. This means any monomial is a
polynomial. For example, two 𝑥 is also a
polynomial because it’s the sum of one monomial. However, we can also create more
polynomials. Let’s add together two 𝑥 to the
first power and negative 𝑦 squared. Adding these two monomials together
means that two 𝑥 to the first power plus negative 𝑦 squared is a monomial. And of course we can simplify
this. We could just write this as two 𝑥
minus 𝑦 squared. This is also an example of a
polynomial.
This is an important example to
illustrate when we say a polynomial is the sum of monomials. This does not mean all of our
operations need to be addition, since we know that two 𝑥 minus 𝑦 squared is a
polynomial. We can create more examples of
polynomials. For example, we could add together
a term with 𝑥 with a constant. For example, 𝑥 plus three is an
example of a polynomial. These, in fact, have a special
name. They’re called linear expressions,
because if we plot them on a graph, they make straight lines.
But we don’t need to stop
there. We could add even more monomials
involving 𝑥 to this. For example, we could add two 𝑥
squared to this. This gives us two 𝑥 squared plus
𝑥 plus three is also a polynomial. And we can give one more example of
a polynomial. An expression like negative
one-half multiplied by 𝑧 is a polynomial. This is because it’s a
monomial. So, to check if an expression is a
polynomial, we just look at each part individually and check if it’s a monomial.
So, let’s look at a few examples of
expressions which are not polynomials. The first example of an expression
which is not a polynomial is 𝑥 to the power of negative two. And the reason for this is for this
to be a polynomial, 𝑥 to the power of negative two must be a monomial. And remember, in a monomial, all of
our variables must have nonnegative integer exponents. However, in our case, the exponent
of 𝑥 is negative two. This is negative, so this is not a
monomial. And hence, this expression is not a
polynomial.
We can use the same reasoning to
come up with more examples which are not polynomials, for example, 𝑥 to the power
of one-half. Once again, for this to be a
polynomial, 𝑥 to the power of one-half needs to be a monomial. But remember, the exponent of 𝑥
needs to be a nonnegative integer. In this case, it’s one-half. This is not an integer, so this is
not a monomial. And hence, this is not a
polynomial.
And we know something about 𝑥 to
the power of one-half. By using our laws of exponents, we
can rewrite this as the square root of 𝑥. So, we also know the square root of
𝑥 is not a polynomial because the exponent of 𝑥 is not an integer. But then, if we’re allowed to use
our laws of exponents, we can do exactly the same for 𝑥 to the power of negative
two. Remember, raising a number to the
power of negative two is the same as dividing by this raised to the positive
exponent. So, one over 𝑥 squared is also not
a polynomial. The exponent of our variable is
negative.
We’ll give one last example of
something which is not a polynomial. Consider the expression three plus
𝑥𝑦 minus six times 𝑥 to the fourth power multiplied by 𝑦 to the power of
negative one times 𝑧 plus the square root of two. Remember, for this to be a
polynomial, it must be the sum of monomials. So, we’ll check each individual
part to check if it’s a monomial. We’ll start with three. This is a constant, so it’s a
monomial. Next, we have 𝑥 multiplied by
𝑦. Remember, 𝑥 is equal to 𝑥 to the
first power, and 𝑦 is equal to 𝑦 to the first power. So, the exponent of 𝑥 and the
exponent of 𝑦 are nonnegative integers. Therefore, 𝑥 times 𝑦 is also a
monomial.
However, we now see we have a
problem. We have 𝑦 raised to the power of
negative one. And remember, in our monomials, our
variables are not allowed to have negative exponents. Therefore, this expression is not a
polynomial because one of the variables has a negative exponent.
Before we continue, it’s also worth
pointing out we often call each individual part of an expression a term. So, for example, in our most recent
example, it contains four terms.
Now that we’ve done all of this,
we’re ready to define a couple of key properties which will help us describe certain
attributes of polynomials. First, we’ll define what we mean by
the degree of a polynomial. The degree of a polynomial is the
greatest sum of the exponents of our variables in a single term. This is a very complicated-sounding
definition. However, it’s easier if we go
through a few examples.
Let’s start by finding the degree
of a few polynomials we’ve already found. We’ll start with two 𝑥. First, we’ll need to look term by
term. Well, this polynomial only has one
term. So, we can just focus on two
𝑥. Next, we need to find the sum of
the exponents of the variables in this term. To do this, we’ve already seen that
we can write 𝑥 as 𝑥 to the first power. So, in fact, this only has one
variable, and its exponent is one. So, we say the degree of two 𝑥 is
one.
Another example we could look at is
the constant three. Remember, this is an example of a
polynomial. And at first, it might seem tricky
how we can find the degree of this polynomial. Since it’s a constant, it doesn’t
contain any variables. So, how are we supposed to find the
sum of the exponents of all of the variables? Luckily, by using our laws of
exponents, we know that 𝑥 to the zeroth power is also equal to one. So, we can rewrite three as three
times 𝑥 to the zeroth power. Then, just like before, we can see
that the degree of this polynomial will be zero. Therefore, we were able to show the
degree of this polynomial is zero. In fact, the exact same is true for
any constant. If we consider it as a polynomial,
then the degree of a constant polynomial is always equal to zero.
Let’s now try finding the degree of
a polynomial with more than one term. Let’s try and find the degree of
two 𝑥 squared plus 𝑥 plus three. Once again, remember, when we’re
looking for the degree of a polynomial, we’re only interested in the greatest sum of
exponents of the variables in a single term. So, we can look at the degree of
each term separately. So, let’s start with the first term
in this polynomial. That’s two 𝑥 squared. This term only contains one
variable. And we can see its exponent is
two. So, the degree of two 𝑥 squared is
equal to two.
Now, let’s look at our second
term. Well, we can see it’s equal to
𝑥. And we’ve already seen that we can
write this as 𝑥 to the first power. So, its degree is one. Finally, we have our third and
final term. It’s a constant, so its degree is
equal to zero. Therefore, the degree of this
polynomial will be the largest of these values. Its degree will be two. Therefore, we were able to show the
polynomial two 𝑥 squared plus 𝑥 plus three has degree two.
But so far, we’ve only seen how to
find the degree of polynomials where each term only contains one variable. What if we were to try and find the
degree of root two times 𝑥 times 𝑦 squared? Remember, we can do this term by
term, and we need to find the sum of the exponents of our variables. So, once again, we’ll write 𝑥 as
𝑥 to the first power. Then, we add the exponents of our
variables together. We get one plus two, which is of
course equal to three. Therefore, the polynomial root two
times 𝑥 times 𝑦 squared has degree three.
There’s one more thing we need to
define before we move on to answering some questions. We want to define the constant
factor of a term as the coefficient of that term. Another way of saying this is the
coefficient is the numerical factor in an algebraic term. This usually appears at the start
of our term. For example, in the term two 𝑥, we
have the coefficient is two. It’s the constant factor of this
term. Similarly, if we look at the term
negative 𝑦 squared, then the coefficient of negative 𝑦 squared is negative
one. Saying the coefficient of a term
gives us a nice way to explain the part of the term which doesn’t vary as our
variables change.
Let’s now see some examples of how
we’d use all of this to answer some questions.
Which of the following expressions
are polynomials? Expression (A) 𝑥 squared plus five
𝑥𝑦 minus two. Expression (B) 𝑥 cubed times 𝑦
squared. Expression (C) 𝑥 to the power of
negative one times 𝑦 to the fourth power. Expression (D) five over 𝑥 minus
four 𝑥𝑦.
To answer this question, we first
need to recall that polynomials are the sum of monomials. And remember, monomials are the
products of constants and variables raised to nonnegative integer exponents. So, to check if these four
expressions are polynomials, we need to see if any of our variables are raised to
nonnegative integer exponents.
If we start with expression (A), we
can see this is indeed the sum of monomials. All of our variables are raised to
positive integer values. So, expression (A) is the sum of
monomials. Therefore, it’s a polynomial. And the exact same is true for
expression (B). The exponents three and two are
both positive integers.
However, in expression (C), we can
see the exponent of 𝑥 is negative one. And if one of our variables
contains a negative exponent, then this is not a monomial. So, expression (C) is also not a
polynomial. We can see something very similar
is true for expression (D). By using our laws of exponents, we
know dividing by 𝑥 is the same as multiplying by 𝑥 to the power of negative
one. But that means in this term we have
a negative exponent of our variable 𝑥. So, five over 𝑥 is not a
monomial. Therefore, expression (D) is not
the sum of monomials, so it’s not a polynomial.
Therefore, we were able to show, of
the four given expressions, only expressions (A) and (B) were polynomials.
Let’s now see an example of
determining the degree of a polynomial.
Determine the degree of 𝑦 to the
fourth power minus seven 𝑦 squared.
In this question, we’re asked to
find the degree of an algebraic expression. And we can see something
interesting about this expression. All of our variables are raised to
positive integer values. In other words, this expression is
the sum of monomials. So, it’s a polynomial. So, we’re asked to find the degree
of a polynomial. To do this, let’s start by
recalling what we mean by the degree of a polynomial.
We recall the degree of a
polynomial is the greatest sum of the exponents of the variables in any single
term. What this means is we look at each
individual term, we add together all of the exponents of our variables, and we want
to find the biggest value that this gives us. So, let’s start with the first term
in our expression, 𝑦 to the fourth power.
In this case, there’s only one
variable and its exponent is four, so the degree of 𝑦 to the fourth power is
four. Next, let’s look at our second
term, negative seven 𝑦 squared. Once again, there’s only one
variable, and we can see its exponent. Its exponent is two. So, the degree of negative seven 𝑦
squared is equal to two. And the degree of our polynomial is
the biggest of these numbers. Therefore, its degree is four.
And in fact, we can use the exact
same method to find the degree of any polynomial with only one variable. Its degree will just be the highest
exponent of that variable which appears in our polynomial. Therefore, we were able to show 𝑦
to the fourth power minus seven 𝑦 squared is a fourth-degree polynomial.
Let’s now see an example of finding
the degree and coefficient of a monomial.
Determine the coefficient and the
degree of negative seven 𝑥 cubed.
We’re given an algebraic
expression, negative seven 𝑥 cubed, and we’re asked to find the coefficient and
degree of this expression. First, we can see that this only
contains one term. And we can see that our variable 𝑥
is raised to the power of three. Because this is a positive integer,
this means this is an example of a monomial or a polynomial. So, let’s start by recalling what
we mean by the coefficient of a monomial. This means the numerical factor of
our monomial. In our case, we can see the
numerical factor is negative seven. So, the coefficient of this
monomial is negative seven.
Let’s now find the degree of this
monomial. We could write out the full
definition of the degree of a polynomial. However, we notice that our
monomial only contains one variable. So, in actual fact, there’s an
easier way to find the degree. When our polynomial only contains
one variable, the degree of this polynomial will always be the highest exponent of
that variable which appears. And in our case, we only have one
instance of our variable. And its exponent is three because
we have 𝑥 cubed. Therefore, we were able to show the
coefficient of negative seven 𝑥 cubed is negative seven and the degree of negative
seven 𝑥 cubed is three.
Let’s now see how we can use the
definition of a degree to find the value of a constant.
If the degree of seven 𝑥 to the
fifth power is the same as that of negative six 𝑦 to the power of 𝑛, what is the
value of 𝑛?
We’re given two algebraic
expressions. And we’re told that both of these
have the same degree. To answer this question, we first
need to find the degree of seven 𝑥 to the fifth power. We can see this is a monomial
because it’s one term and the exponent of 𝑥 is five, which is a positive
integer. Now, we could use the fact that
this is a polynomial to find the degree of this expression as a polynomial. However, we can also use an
equivalent definition because our expression only contains one term.
The degree of an algebraic term is
the sum of the exponents of all the variables in that term. And this will give us the same
answer as the degree of our polynomial because this has only one term. We can see the exponent of our
variable is five. Therefore, seven 𝑥 to the fifth
power is of degree five. But then, the question tells us
that negative six 𝑦 to the 𝑛th power has the same degree. So, it must also be of degree
five. And then, because this is also a
singular term, the sum of all the exponents of the variables must be equal to
five.
But we can see there’s only one
exponent on our variable, the unknown 𝑛. Therefore, to make seven 𝑥 to the
fifth power and negative six 𝑦 to the 𝑛th of power have the same degree, we must
have that 𝑛 is equal to five.
Let’s now see an example of finding
the number of terms in an algebraic expression.
How many terms are in the
expression four 𝑥 minus 𝑦 squared plus 27?
To answer this question, we first
need to recall what we mean by a term. And in mathematics, terms is one of
those words which has many different definitions depending on the context. In this context, we’re asked for
the number of terms in an algebraic expression. And this can mean one of two
things. It could be the number of monomials
in our expression, or it could also be the number of like expressions in our
expression. In this case, both of these will
give us the same answer. We’ll just use the number of
monomials in our expression.
Remember, a monomial is the product
between constants and variables raised to the power of nonnegative integers. We can see in our case there are
three monomials in this expression. Four 𝑥 is a monomial because we
can write 𝑥 as 𝑥 to the first power. Negative 𝑦 squared is a monomial
because negative one is a constant and two is a nonnegative integer. Finally, 27 is a monomial because
27 is a constant. Therefore, because our expression
contained three different monomials, we were able to show that there were three
terms in this expression.
Sometimes, we might also be asked
to pick out specific information about our polynomial. Let’s see an example of this.
What is the constant term in the
expression four 𝑥 minus 𝑦 squared plus 27?
To answer this question, we first
need to recall what we mean by the constant term in an expression. First, a constant is something
whose value does not change. For example, in our expression, we
call 𝑥 and 𝑦 variables because they can take on many different values. However, a number like 27 doesn’t
change as our values of 𝑥 or 𝑦 change. It’s always equal to 27.
Next, we also need to remember what
we mean by a term. In this context, when we say a
term, we mean the parts we’re adding together to make our expression. So, four 𝑥 is a term, negative 𝑦
squared is a term, and 27 is a term. Alternatively, we can think of each
monomial as a term. We can see that four 𝑥 is varying
as the value of 𝑥 changes and 𝑦 squared is changing as the value of 𝑦
changes. So, only 27 remains constant. Therefore, the constant term in the
expression four 𝑥 minus 𝑦 squared plus 27 is 27.
Let’s now go over the key points of
this video. First, we defined a polynomial to
be the sum of one or more monomial. And remember, a monomial is the
product between constants and variables, where our variables are all raised to
nonnegative exponents. Next, we also defined the degree of
a polynomial. The degree of a polynomial is the
greatest sum of the exponents of the variables in any term in our polynomial.
And this gave us two useful
results. First, if our polynomial only has
one term, then we can just sum the exponents of the variables in this term to find
its degree. And we also saw if our polynomial
only contains one variable, then to find its degree, all we need to do is find the
highest exponent of this variable which appears in our polynomial.
Finally, we defined the coefficient
of a term to be the numerical factor of that term. Another way of thinking of this is
the coefficient of a term is the number used to multiply any of the variables. And we know this is usually written
at the front of our term to avoid confusion. But it’s not always written at the
front. For example, the term 𝑥 over two
has the coefficient of one-half because we’re multiplying 𝑥 by one-half.
