Given P x (Select all that apply.) 4x³ + 11x² + 16x − 9, which of the following is true? = As x → -∞, P(x) → ∞. As x → ∞, P(x) → ∞. As x → ∞, P(x) →-80. As x → -∞, P(x) - →-80.
Given P x (Select all that apply.) 4x³ + 11x² + 16x − 9, which of the following is true? = As x → -∞, P(x) → ∞. As x → ∞, P(x) → ∞. As x → ∞, P(x) →-80. As x → -∞, P(x) - →-80.
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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![### Polynomial Behavior at Infinity
Given \( P(x) = 4x^3 + 11x^2 + 16x - 9 \), which of the following is true?
(Select all that apply.)
- [ ] As \( x \rightarrow -\infty \), \( P(x) \rightarrow \infty \).
- [ ] As \( x \rightarrow \infty \), \( P(x) \rightarrow \infty \).
- [ ] As \( x \rightarrow \infty \), \( P(x) \rightarrow -\infty \).
- [ ] As \( x \rightarrow -\infty \), \( P(x) \rightarrow -\infty \).
### Explanation:
You are given a polynomial function \( P(x) = 4x^3 + 11x^2 + 16x - 9 \). To determine the behavior of the polynomial as \( x \) approaches positive or negative infinity, we focus on the term with the highest degree, which in this case is \( 4x^3 \). The sign of the leading coefficient and the degree of the term determine the end behavior:
- For \( x \rightarrow \infty \): Since the leading term \( 4x^3 \) is positive and cubic, \( P(x) \) will tend towards \( \infty \).
- For \( x \rightarrow -\infty \): The leading term \( 4x^3 \) dominates and because it’s a cubic term with a positive coefficient, \( P(x) \) will tend towards \( -\infty \).
Therefore, the correct statements are:
- As \( x \rightarrow -\infty \), \( P(x) \rightarrow -\infty \).
- As \( x \rightarrow \infty \), \( P(x) \rightarrow \infty \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd66c9551-2207-4a1f-96cc-82c211723c9b%2F69d188c3-3b4c-4e26-9701-9cc287f6c5bf%2F2jnuaz_processed.png&w=3840&q=75)
Transcribed Image Text:### Polynomial Behavior at Infinity
Given \( P(x) = 4x^3 + 11x^2 + 16x - 9 \), which of the following is true?
(Select all that apply.)
- [ ] As \( x \rightarrow -\infty \), \( P(x) \rightarrow \infty \).
- [ ] As \( x \rightarrow \infty \), \( P(x) \rightarrow \infty \).
- [ ] As \( x \rightarrow \infty \), \( P(x) \rightarrow -\infty \).
- [ ] As \( x \rightarrow -\infty \), \( P(x) \rightarrow -\infty \).
### Explanation:
You are given a polynomial function \( P(x) = 4x^3 + 11x^2 + 16x - 9 \). To determine the behavior of the polynomial as \( x \) approaches positive or negative infinity, we focus on the term with the highest degree, which in this case is \( 4x^3 \). The sign of the leading coefficient and the degree of the term determine the end behavior:
- For \( x \rightarrow \infty \): Since the leading term \( 4x^3 \) is positive and cubic, \( P(x) \) will tend towards \( \infty \).
- For \( x \rightarrow -\infty \): The leading term \( 4x^3 \) dominates and because it’s a cubic term with a positive coefficient, \( P(x) \) will tend towards \( -\infty \).
Therefore, the correct statements are:
- As \( x \rightarrow -\infty \), \( P(x) \rightarrow -\infty \).
- As \( x \rightarrow \infty \), \( P(x) \rightarrow \infty \).
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