### 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.