Lesson Video: Degree and Coefficient of Polynomials Mathematics

In this video, we will learn how to determine the degree of a polynomial and use the terminology associated with polynomials, such as terms, coefficients, and constants.

17:11

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.

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