### Video Transcript

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

Letβs 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 π₯ as
π₯ 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.

Letβs 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 π₯ 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 π₯. 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 π₯π¦ raised to the fifth power has the highest degree, which is
option (B).