c. Determine the degree, the leading coefficient, the constant term, the roots and the number of
real roots of the following polynomial equations.
Answers
Degree:
8
Leading term:
3
x
5
Leading Coefficient:
3
Constant:
1
End behavior: See below in blue
Explanation:
The degree is the sum of the exponents on all terms. Our exponents are
5
,
2
and
1
, which sum up to
8
. This is the degree of our polynomial
g
(
x
)
.
The leading term of a polynomial is just the term with the highest degree, and we see this is
3
x
5
.
The leading coefficient is just the number multiplying the highest degree term. The coefficient on
3
x
5
is
3
.
The constant term is just a term without a variable. In our case, the constant is
1
.
For end behavior, we want to consider what our function goes to as
x
approaches positive and negative infinity.
In our polynomial
g
(
x
)
, the term with the highest degree is what will dominate the end behavior. So let's take the limit of it:
lim
x
→
∞
3
x
5
=
∞
lim
x
→
−
∞
3
x
5
=
−
∞
Our limits describe our limit behavior.
Hope this helps!