### Problem 5: Finding Asymptotic Behavior of a Polynomial**Problem Statement:**Find the asymptotic behavior of the polynomial $ f(x) = -x^7 + 4x^6 + 2x^5 - x^3 + 10x^2 - 12x - 10 $ (i.e., determine the behavior of $ f(x) $ as $ x \to \pm \infty $).---**Detailed Explanation:**Understanding the asymptotic behavior of a polynomial function involves analyzing the behavior of the function as the variable $ x $ approaches positive and negative infinity. In this problem, the given polynomial is:\[ f(x) = -x^7 + 4x^6 + 2x^5 - x^3 + 10x^2 - 12x - 10 \]To determine the behavior of $ f(x) $ as $ x \to \pm \infty $, we need to focus on the term with the highest power of $ x $. In this polynomial, the highest power of $ x $ is $ x^7 $.Since the coefficient of $ x^7 $ is negative ( $-1$ ), the term $ -x^7 $ will dominate the behavior of $ f(x) $ as $ x $ grows large in magnitude (both positively and negatively).- As $ x \to \infty $: \[ -x^7 \to -\infty \] Therefore, $ f(x) \to -\infty $.- As $ x \to -\infty $: \[ -x^7 \to -(-\infty)^7 = -(-\infty) = \infty \] Therefore, $ f(x) \to \infty $.In conclusion, the polynomial $ f(x) $ behaves as follows asymptotically:- $ f(x) \to -\infty $ as $ x \to \infty $- $ f(x) \to \infty $ as $ x \to -\infty $This analysis highlights that for polynomials, the term with the highest exponent significantly influences the function's behavior at extreme values of $ x $. In this case, the dominant term $ -x^7 $ leads to the

Find the degree and the leading coefficient for the given polynomial.

Answered: . Find the asymptotic behavior of the…

. Find the asymptotic behavior of the polynomial f(x) = -x² + 4x® + 2x5 12x – 10 (i.e. determine the behavior of f (x) as x →±∞x). – 2x3 + 10x² -

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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Question

### Problem 5: Finding Asymptotic Behavior of a Polynomial

**Problem Statement:**

Find the asymptotic behavior of the polynomial $ f(x) = -x^7 + 4x^6 + 2x^5 - x^3 + 10x^2 - 12x - 10 $ (i.e., determine the behavior of $ f(x) $ as $ x \to \pm \infty $).

---

**Detailed Explanation:**

Understanding the asymptotic behavior of a polynomial function involves analyzing the behavior of the function as the variable $ x $ approaches positive and negative infinity.

In this problem, the given polynomial is:

\[ f(x) = -x^7 + 4x^6 + 2x^5 - x^3 + 10x^2 - 12x - 10 \]

To determine the behavior of $ f(x) $ as $ x \to \pm \infty $, we need to focus on the term with the highest power of $ x $. In this polynomial, the highest power of $ x $ is $ x^7 $.

Since the coefficient of $ x^7 $ is negative ( $-1$ ), the term $ -x^7 $ will dominate the behavior of $ f(x) $ as $ x $ grows large in magnitude (both positively and negatively).

- As $ x \to \infty $:
\[ -x^7 \to -\infty \]
Therefore, $ f(x) \to -\infty $.

- As $ x \to -\infty $:
\[ -x^7 \to -(-\infty)^7 = -(-\infty) = \infty \]
Therefore, $ f(x) \to \infty $.

In conclusion, the polynomial $ f(x) $ behaves as follows asymptotically:
- $ f(x) \to -\infty $ as $ x \to \infty $
- $ f(x) \to \infty $ as $ x \to -\infty $

This analysis highlights that for polynomials, the term with the highest exponent significantly influences the function's behavior at extreme values of $ x $. In this case, the dominant term $ -x^7 $ leads to the

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