Question
Find the degree, leading term, leading coefficient, constant term, and end behavior of the
given polynomial.
Degree:
Leading term:
Leading coefficient:
End behavior:
As
As
given polynomial.
Degree:
Leading term:
Leading coefficient:
End behavior:
As
As
Ask by Weston Wood. in the United States
Jan 23,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
Degree: 8
Leading term:
Leading coefficient: -5
Constant term: 0
End behavior:
Leading term:
Leading coefficient: -5
Constant term: 0
End behavior:
- As
- As
Solution
Let’s analyze the polynomial
to determine its degree, leading term, leading coefficient, constant term, and end behavior.
1. Degree
The degree of a polynomial is the highest power of the variable
after expanding the polynomial.
-
has a degree of 5. -
has a degree of 1. -
has a degree of 2.
Total Degree:
.
Degree: 8
2. Leading Term and Leading Coefficient
The leading term is found by multiplying the leading terms of each factor:
- Leading term of
is . - Leading term of
is . - Leading term of
is .
Leading Term Calculation:
Leading Term:
The leading coefficient is the coefficient of the leading term, which is -5.
Leading Coefficient: -5
3. Constant Term
The constant term is the term in the polynomial that does not contain the variable
.
When expanding
, every term will have a power of
higher than or equal to 5. Therefore, there is no constant (degree 0) term in this polynomial.
Constant Term: 0
4. End Behavior
The end behavior of a polynomial is determined by its leading term.
- Leading Term:
- Since the degree is even and the leading coefficient is negative, as
approaches both and , the polynomial will approach .
- Since the degree is even and the leading coefficient is negative, as
End Behavior:
- As
- As
Summary
- Degree: 8
- Leading term:
- Leading coefficient: -5
- Constant term: 0
- End behavior:
- As
- As
- As
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg


Mind Expander
To analyze the polynomial
, we can start by expanding it step by step.
First, find the degree of each factor:
- The first factor
has a degree of 5. - The second factor
has a degree of 1. - The third factor
has a degree of 2.
When finding the overall degree of the polynomial
, we add the degrees together:
Now let’s look for the leading term and leading coefficient. The leading term is determined by multiplying the leading terms of each factor:
- The leading term from
is . - The leading term from
is . - The leading term from
is .
Thus, the overall leading term is:
The leading coefficient, which is the coefficient of the leading term, is
.
The constant term is the value of the polynomial when
:
Finally, regarding end behavior, a polynomial with an even degree and a negative leading coefficient (like
) will trend downward on both ends:
- As
- As
In summary:
Degree: 8
Leading term:
Leading coefficient:
Constant term: 0
As
As
Degree: 8
Leading term:
Leading coefficient:
Constant term: 0
As
As