Question
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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

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:
  • 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 .
End Behavior:
  • As
  • As

Summary

  • Degree: 8
  • Leading term:
  • Leading coefficient: -5
  • Constant term: 0
  • End behavior:
    • As
    • As

Answered by UpStudy AI and reviewed by a Professional Tutor

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Explain
Simplify this solution

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

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