Algebra: Structure And Method, Book 1
(REV)00th Edition
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Chapter4: Polynomials
Section4.3: Multiplying Monomials
Problem 20OE
Related questions
Question
![**Determine the end behavior for the following polynomial**
\[ -3x^5 + 2x^3 - x + 2 \]
**Multiple Choice Options:**
1. Option A:
- ⃝ \( \uparrow \; \uparrow \)
2. Option B:
- ⃝ \( \downarrow \; \downarrow \)
3. Option C:
- ⃝ \( \downarrow \; \uparrow \)
4. Option D:
- ⃝ \( \uparrow \; \downarrow \)
### Explanation:
The end behavior of a polynomial is determined by its leading term. The given polynomial is:
\[ -3x^5 + 2x^3 - x + 2 \]
The leading term is the term with the highest power of \(x\), which in this case is \(-3x^5\).
For polynomials, the end behavior is dictated by the degree (the highest exponent) and the leading coefficient (the coefficient of the term with the highest power of \(x\)):
- If the degree is odd and the leading coefficient is negative, the polynomial will start at \( \uparrow \) (approaching negative infinity on the left side) and end at \( \downarrow \) (approaching positive infinity on the right side).
- For this polynomial (\(-3x^5\)), with an odd degree (5) and a negative leading coefficient (-3), the end behavior is \( \uparrow \; \downarrow \).
Therefore, the correct answer is **Option D: \( \uparrow \; \downarrow \)**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8a5ce0a8-2e23-4118-8872-76b2d5c7da77%2F234ec7ec-86f7-482d-bfe3-74248870e795%2Fyvkq3jl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Determine the end behavior for the following polynomial**
\[ -3x^5 + 2x^3 - x + 2 \]
**Multiple Choice Options:**
1. Option A:
- ⃝ \( \uparrow \; \uparrow \)
2. Option B:
- ⃝ \( \downarrow \; \downarrow \)
3. Option C:
- ⃝ \( \downarrow \; \uparrow \)
4. Option D:
- ⃝ \( \uparrow \; \downarrow \)
### Explanation:
The end behavior of a polynomial is determined by its leading term. The given polynomial is:
\[ -3x^5 + 2x^3 - x + 2 \]
The leading term is the term with the highest power of \(x\), which in this case is \(-3x^5\).
For polynomials, the end behavior is dictated by the degree (the highest exponent) and the leading coefficient (the coefficient of the term with the highest power of \(x\)):
- If the degree is odd and the leading coefficient is negative, the polynomial will start at \( \uparrow \) (approaching negative infinity on the left side) and end at \( \downarrow \) (approaching positive infinity on the right side).
- For this polynomial (\(-3x^5\)), with an odd degree (5) and a negative leading coefficient (-3), the end behavior is \( \uparrow \; \downarrow \).
Therefore, the correct answer is **Option D: \( \uparrow \; \downarrow \)**.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Algebra: Structure And Method, Book 1
Algebra
ISBN:
9780395977224
Author:
Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:
McDougal Littell
Elementary Algebra
Algebra
ISBN:
9780998625713
Author:
Lynn Marecek, MaryAnne Anthony-Smith
Publisher:
OpenStax - Rice University
Algebra: Structure And Method, Book 1
Algebra
ISBN:
9780395977224
Author:
Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:
McDougal Littell
Elementary Algebra
Algebra
ISBN:
9780998625713
Author:
Lynn Marecek, MaryAnne Anthony-Smith
Publisher:
OpenStax - Rice University