Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**Determining the End Behavior of a Polynomial**
**Given the following polynomial, determine the end behavior:**
\[ 4x^{12} - 34x^7 + 15x^2 \]
**Explanation of Diagrams:**
This section utilizes diagrams to determine the end behavior of the given polynomial. Each diagram consists of upward and downward pointing arrows, which represent the direction of the polynomial as \( x \) approaches positive and negative infinity.
1. **First Diagram**:
- Two upward pointing arrows (↑ ↑).
2. **Second Diagram**:
- Two downward pointing arrows (↓ ↓).
3. **Third Diagram**:
- Mixed arrows with one downward and one upward arrow (↓ ↑).
4. **Fourth Diagram**:
- Mixed arrows with one upward and one downward arrow (↑ ↓).
**Steps to Determine End Behavior:**
To determine the end behavior of the polynomial \( 4x^{12} - 34x^7 + 15x^2 \):
1. Focus on the leading term, which is \( 4x^{12} \).
2. The degree of the leading term is 12, which is an even number.
3. The coefficient of the leading term is 4, which is positive.
**Conclusion**:
Given the leading term \( 4x^{12} \), the polynomial will exhibit the following end behavior:
- As \( x \) approaches \( +\infty \) (positive infinity), \( 4x^{12} \rightarrow +\infty \).
- As \( x \) approaches \( -\infty \) (negative infinity), \( 4x^{12} \rightarrow +\infty \).
Thus, the correct diagram illustrating the end behavior is the first diagram, which shows two upward pointing arrows (↑ ↑). This indicates that the polynomial rises to positive infinity on both ends.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F15430a79-cfc8-483e-83f7-796f8bb05713%2Fa1362dd8-97dd-4d22-8d49-41625f7010d7%2Fkqhi1w9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Determining the End Behavior of a Polynomial**
**Given the following polynomial, determine the end behavior:**
\[ 4x^{12} - 34x^7 + 15x^2 \]
**Explanation of Diagrams:**
This section utilizes diagrams to determine the end behavior of the given polynomial. Each diagram consists of upward and downward pointing arrows, which represent the direction of the polynomial as \( x \) approaches positive and negative infinity.
1. **First Diagram**:
- Two upward pointing arrows (↑ ↑).
2. **Second Diagram**:
- Two downward pointing arrows (↓ ↓).
3. **Third Diagram**:
- Mixed arrows with one downward and one upward arrow (↓ ↑).
4. **Fourth Diagram**:
- Mixed arrows with one upward and one downward arrow (↑ ↓).
**Steps to Determine End Behavior:**
To determine the end behavior of the polynomial \( 4x^{12} - 34x^7 + 15x^2 \):
1. Focus on the leading term, which is \( 4x^{12} \).
2. The degree of the leading term is 12, which is an even number.
3. The coefficient of the leading term is 4, which is positive.
**Conclusion**:
Given the leading term \( 4x^{12} \), the polynomial will exhibit the following end behavior:
- As \( x \) approaches \( +\infty \) (positive infinity), \( 4x^{12} \rightarrow +\infty \).
- As \( x \) approaches \( -\infty \) (negative infinity), \( 4x^{12} \rightarrow +\infty \).
Thus, the correct diagram illustrating the end behavior is the first diagram, which shows two upward pointing arrows (↑ ↑). This indicates that the polynomial rises to positive infinity on both ends.
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