Which of the following expressions are degree five polynomials? (a) 𝑥 to the fifth power plus five 𝑥 to the sixth power minus two. (b) 𝑥 to the fourth power times 𝑦 minus 𝑥 to the fourth power minus three 𝑥 squared. (c) 𝑥 cubed 𝑦 squared minus four 𝑥𝑦 squared. (d) 𝑥 to the fifth power minus 𝑦 times 𝑥 to the power of negative one. And (e) 𝑥 cubed minus two 𝑥𝑦 plus five 𝑥 to the fifth power.
In this question, we’re asked to determine which of five given expressions are degree five polynomials. So we should start by recalling what we mean by a polynomial expression and what it means for a polynomial to have degree five.
First, we recall a polynomial is the sum of monomials, where a monomial is a product of constants and variables, where our variables must have nonnegative integer exponents. We can use this definition to determine which of the five given expressions are polynomials first.
Let’s start with option (a). To determine if this is a polynomial, we first need to check whether each of the five terms are monomials. This means they need to be the products of constants and variables and the variables must have nonnegative integer exponents. First, we see the only variable is 𝑥 and the exponents of 𝑥 are five and six. These are nonnegative integers. Next, we need to check that each term is the product of constants and variables. Although it’s not necessary, we can rewrite the first term as one times 𝑥 to the fifth power and the third term as negative two 𝑥 to the zeroth power. And if we wanted to be even more careful, we could remember that subtracting two 𝑥 to the zeroth power is the same as adding negative two 𝑥 to the zeroth power. In either case, we can conclude each term is a monomial. So the expression in option (a) is a polynomial.
We get a very similar story in option (b). Each term is the product of constants and variables. However, we can see we do have two different variables 𝑥 and 𝑦. So we do need to be careful. We need to check that the exponents of all of our variables are nonnegative integers. In this case, we just write 𝑦 as 𝑦 to the first power. So the exponents of our variables are four, one, four, and two. These are nonnegative integers.
The same is true in option (c). Each term is the product of constants and variables. And we can see all of the variables are raised to nonnegative integer exponents. However, the same is not true in option (d). We can see we have a term 𝑥 to the power of negative one. This is a variable raised to a negative integer exponent. So (d) does not represent a polynomial. So we can exclude option (d). And we can also see that option (e) is a polynomial.
Now that we’ve concluded expressions (a), (b), (c), and (e) are polynomials, let’s recall how we check the degree of a polynomial. We can recall the degree of any polynomial is the greatest sum of the exponents of the variables which appear in any single term. And it’s worth pointing out we are only interested in nonzero terms. If we have a factor of zero in our term, we can just remove this term and not change the expression. We can use this definition to determine the degree of the four polynomial expressions we’re given.
And to do this, we first note, in the definition of a degree, we take the sum of the exponents of the variables in a single term. This means if we’re working with a single variable polynomial, we don’t need to take a sum because there’s only one variable. In this case, its degree will just be the largest exponent of that variable which appears in a single nonzero term. For example, in expression (a), we can see it’s a single variable polynomial. And we can also see that all of the terms are nonzero. So we can just check which of the exponents of 𝑥 is the highest. And we can see that this is six. So expression (a) is a degree six polynomial. Therefore, expression (a) is not a degree five polynomial. So we can remove this option.
The remaining expressions (b), (c), and (e) are two-variable polynomials. So we are going to need to take the sum of the exponents of the variables in each term. Let’s start with expression (b). We’ll rewrite 𝑦 as 𝑦 to the first power. We need to find the sum of the exponents of the variables in each term. In the first term, the exponent of 𝑥 is four and the exponent of 𝑦 is one. So we have four plus one is equal to five. In the second term, we have a single variable of 𝑥. So we just write this as four. And in the third term, we only have a single variable of 𝑥. So this sum will just give us two. The degree of this polynomial is the biggest of these three values, which we can see is five. And therefore, the degree of (b) is five. So (b) is a degree five polynomial.
We want to do the same for expression (c). We need to add the exponents of the variables in the first term. That’s three plus two, which we can evaluate is five. We can then do the same for the second term. We write 𝑥 as 𝑥 to the first power. Then we add the exponents of the variables in this term. One plus two is three. And then the larger of these two values is the degree of our polynomial. We can see this is also five. So (c) is a degree five polynomial.
Finally, we move on to option (e). The first term is a single variable of 𝑥. So this term has degree three. The second term can be rewritten as negative two times 𝑥 to the first power times 𝑦 to the first power. And one plus one is equal to two. So the second term has degree two. Finally, the third term is in a single variable of 𝑥, so its degree is five. And since all of these are nonzero terms, the degree is the highest of these values. We can see that this is once again five. Therefore, option (e) is also a degree five polynomial.
Therefore, we were able to show, of the five given expressions, only expressions (b), (c), and (e) are degree five polynomials.