We are asked to identify the degree, leading term, and leading coefficient of the following
polynomial functions. We identify the highest power of the variable to find the degree.
We identify the term containing the highest power of the variable to find the leading
term. And finally, we identify the coefficient of the leading term to find the leading coefficient.
So for number 1, we're given the function f of x equals 4 minus 3 x squared minus 5
x to the third. Let's begin by determining the degree of each term. The first term is
4. Any constant term has degree 0. The second term is negative 3 x squared, which has degree
2 given by the exponent of 2 on the variable x. The third term is negative 5 x cubed, which
has degree 3, given by the exponent of 3 on the variable x. So because the highest power
on the variable is 3, this is a degree 3 polynomial function. And because the term containing
the highest power on the variable is negative 5 x cubed, negative 5 x cubed is the leading
term. The coefficient of the leading term is negative 5, and therefore the leading coefficient
is negative 5. Now I do want to mention one more thing. Often a polynomial function is
given in descending order, which means the terms are listed from highest degree to lowest
degree. If we were to write f of x in descending order, we would have f of x equals negative
5 x cubed minus 3 x squared plus 4. One nice thing about having a polynomial in descending
order is that we can determine the degree, the leading term, and the leading coefficient
from the first term of the polynomial function, again as long as its in descending order.
Number 2, we are given g of x equals one half t to the sixth minus t to the fourth plus
3 t squared minus 1. Let's begin by determining the degree of each term. The degree of the
first term is 6 because of the exponent of 6 on the variable t. The degree of the second
term is 4, given by the exponent of 4 on the variable t. The degree of the third term is
2. The degree of the last term is 0. Because the highest power on the variable is 6, this
is a degree 6 polynomial function. Because the term containing the highest power on the
variable is one half t to the sixth, the term one half t to the sixth is the leading term.
And because the coefficient of the leading term is one half, the leading coefficient
is one half. Notice this time, g of t was given to us in descending order. Number 3,
we have h of s equals s, or if we want s to the first, minus s to the fourth minus 5.
The degree of the first term is 1, given by the exponent of 1 on the variable s. The degree
of the second term is 4, given by the exponent of 4 on the s, and the degree of the constant
term, negative 5, is 0. Because the highest power on the variable is 4, this is a degree
4 polynomial function. Because the term containing the highest power of the variable is negative
s to the fourth, negative s to the fourth is the leading term. Notice how we have a
negative here because of the minus s to the fourth. Remember minus s to the fourth is
the same as plus negative s to the fourth. And because negative s to the fourth is equal
to negative 1 s to the fourth, the coefficient of the leading term is negative 1, which means
the leading coefficient is negative 1. So while we are done, let's go ahead and give
h of s in descending order. In descending order h of s is equal to negative s to the
fourth plus s minus 5. I hope you found this helpful.