### 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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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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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