Question Video: Determining the Degree of an Algebraic Expression Mathematics

Video Transcript

Which of the expressions below is of the same degree as three 𝑥 to the eighth power plus three 𝑥 to the fourth power times 𝑦 squared plus four 𝑦 squared? Is it option (A) two 𝑥 to the fourth power plus two 𝑥 to the eighth power times 𝑦 cubed plus three 𝑦 to the fourth power? Is it option (B) three 𝑎 to the seventh power plus three 𝑎 cubed times 𝑏 to the fourth power plus two 𝑏 squared? Is it option (C) three 𝑥 squared plus two 𝑥 to the fourth power times 𝑦 to the fourth power plus three 𝑦 to the seventh power? Or is it option (D) three 𝑏 to the ninth power plus three 𝑎 cubed times 𝑏 plus two 𝑎 to the sixth power?

In this question, we’re given four expressions. And we need to determine which of these four expressions has the same degree as a given polynomial. To answer this question, let’s start by recalling what we mean by the degree of a polynomial expression. It’s the greatest sum of the exponents of the variables in any single term. And we know we can only find the degree of a polynomial expression. This means we need to check two things. First, we need to check whether the four given options are in fact polynomial expressions. Then, we also need to check their degrees.

So let’s start by checking whether the four given options are polynomials. To do this, we’re going to first need to recall what we mean by a polynomial expression. We recall a polynomial expression is one where every term is the product of constants and variables, where our variables must be raised to nonnegative integer exponents. And we can see this is true for the four given options. All of the terms are products of constants and variables, and all of the variables are raised to nonnegative integer exponents. Therefore, all four of the given options are polynomials, and we can calculate their degrees.

To compare the degrees, we’re going to need to first find the degree of the polynomial given to us in the question. And we’re going to do this by finding the degree of each term individually. Let’s start with the first term. We want to find the degree of three 𝑥 to the eighth power. And we can do this from the definition of a degree. It’s the greatest sum of the exponents of the variables in a single term. However, we’re only interested in a single term. So we just need to find the sum of the exponents of the variables in this term. And this term only has one variable 𝑥. So the sum of the exponents of the variables in this term is just going to be one term, eight. This means the degree of this first term is just equal to eight.

We can follow the exact same process to find the degree of the second term. Once again, the degree of this term is the sum of the exponents of the variables. This time, this term has two variables, 𝑥 and 𝑦. So we need to add their exponents together. Four plus two is equal to six. Therefore, the second term in this expression has degree six. We need to apply this process one more time to determine the degree of the third term, four 𝑦 squared. Once again, this term only has one variable, 𝑦. So just like the first term, the degree of this polynomial is just going to be the exponent of this variable. Since the exponent is two, its degree is two.

Finally, remember, the degree of the entire expression is the greatest sum of the exponents of the variables in any single term. And this is another way of saying the degree of the polynomial is the greatest degree of any of its individual terms. And we’ve shown the largest of these three degrees is equal to eight. This means we’ve shown the polynomial expression we’re given in the question has degree eight. And we need to determine which of the four given options has degree eight. We can do this by following the same process for all four of the given options to determine their degree and seeing which one has degree eight.

However, there are a few shortcuts we can use to make this process easier. For example, in option (D), we can notice there’s a variable raised to the ninth power. And this means the degree of this individual term must be greater than or equal to nine. Of course, we can’t just calculate the degree of this term. Since there’s only one variable 𝑏, the degree of this term will just be the power of 𝑏. It will be nine. But we can then notice something interesting. The degree of a polynomial is the greatest degree of any of its individual terms. So because we’ve shown the degree of one of the terms is nine, we can conclude the degree of this polynomial expression must be greater than or equal to nine. This is a lower bound on the degree. Therefore, the degree of option (D) is greater than or equal to nine. It can’t be equal to eight. So option (D) is not correct.

Well, of course, it is worth pointing out we can just calculate the degree of this polynomial. The degree of the second term is the sum of the exponents of the variables: three plus one is four. And the third and final term only has one variable. So its degree is just the exponent of this variable, six. Then, the greatest of these three degrees is nine. So this polynomial has degree nine. If we check all of the other options, we can see that none of the variables are raised to an exponent greater than or equal to nine. So we can’t use this to eliminate any more options. However, we can just find the degree of all three of the remaining options.

Let’s start with option (A). The first and last term are only single-variable terms. So the degree of each of these two terms will just be the exponent of the variable. Both of these will be degree four. Let’s now determine the degree of the second term, two 𝑥 to the eighth power times 𝑦 cubed. This term has two variables, so we need to add the exponents of the variables together. Eight plus three is equal to 11. Then, we can see the greatest of the degrees of any single term is 11. Therefore, we’ve shown the polynomial in option (A) has degree 11. This is not the same degree as the polynomial expression given in the question.

We can apply this exact same process to find the degrees of options (B) and (C). First, in options (B) and (C), the first and last term both have one variable. So the degrees of these terms will be the exponents of these variables, seven and two, respectively. Next, the second term has two variables. So we need to add the exponents of the variables together. Three plus four is equal to seven. We now need to find the largest of these degrees. Both the first and second term have degree seven, which is the largest of the values. So this polynomial has degree seven. This is not the same degree as the polynomial given in the question. So option (B) is not correct.

Now let’s do the exact same process to option (C). Once again, the first and last terms are single variables. So we can find their degree by just looking at the exponent of the variables. The degree of the first term is two, and the degree of the third term is seven. And once again, since the second term has two variables, we need to find the sum of the exponents of the variables to determine the degree of the second term. We have that four plus four is equal to eight, which is the largest degree of the three terms. Therefore, the polynomial expression in option (C) has degree eight. And we can see this is the same degree as the polynomial given to us in the question.

Therefore, we’ve shown, of the four given options, only option (C), three 𝑥 squared plus two 𝑥 to the fourth power times 𝑦 to the fourth power plus three 𝑦 squared, has the same degree as three 𝑥 to the eighth power plus three 𝑥 to the fourth power times 𝑦 squared plus four 𝑦 squared. Both of these polynomials are of degree eight.