Question Video: Identifying the Polynomial with the Highest Degree Mathematics • First Year of Preparatory School

Video Transcript

Which of the following polynomials has the highest degree? Option (A) 𝑦 raised to the fourth power. Option (B) 𝑥𝑦 raised to the fifth power. Option (C) 𝑥 cubed 𝑦 plus three. Option (D) 𝑥 cubed plus three 𝑥 squared minus two 𝑥. Or is it option (E) 𝑎 raised to the fourth power 𝑏 plus 𝑏 raised to the fifth power?

In this question, we are given five polynomials and asked to determine which of the polynomials has the highest degree. To answer this question, we can begin by recalling that the degree of a polynomial is the largest sum of the exponents of the variables in a single term. We can use this definition to find the degree of each of the given polynomials.

Let’s start by finding the degree of the polynomial in option (A). We can see that this polynomial only has a single term, so it is a monomial. It also only has a single variable, so its degree is the exponent of the variable, which is four.

Next, we can see that option (B) also has only a single term, so it is a monomial. We want to find the sum of the exponents of the variables in this term. So we first need to rewrite 𝑥 as 𝑥 raised to the first power. We can then calculate that the sum of the exponents of the variables in this term is one plus five, which equals six. So this monomial has degree six.

The remaining options are not monomials. So we will need to find the degree of each term and then find the greatest of these degrees in order to find the degree of the polynomial.

Let’s start with the sum of the exponents of the variables in the first term of the expression in option (C). We have three plus one equals four. So this term is of degree four. We can then recall that nonzero constants have degree zero, we can see this by rewriting the term as three 𝑥 raised to the zeroth power. The largest of these degrees is four, so we can say that this polynomial has degree four.

In option (D), we can see that each term only has a single variable. So the degree of each term is the power of 𝑥. We can then note that the first term has degree three, the second term has degree two, and the third term has degree one. The greatest of these degrees is three, so the degree of this polynomial is three.

Finally, we can apply this same process to the polynomial in option (E). We see that the sum of the exponents of the variables in the first term is five. And the second term only has a single variable raised to the fifth power, so its degree is five. Hence, the degree of this entire polynomial is five.

Therefore, of the given options, we have shown that 𝑥𝑦 raised to the fifth power has the highest degree, which is option (B).