**Task: Finding the Zeros and Graphing a Polynomial**Given the polynomial equation:\[ y = -x^5 - x^4 + 2x^3 + 2x^2 - x - 1 \]**Objective:**1. **Find the Zeros:** - Determine the values of $ x $ for which $ y = 0 $. - Use techniques such as synthetic division, factoring, or the Rational Root Theorem where applicable.2. **Graph the Polynomial:** - Plot the function on a coordinate plane. - Identify key features such as intercepts, turning points, and end behavior. - Analyze the degree and leading coefficient to determine the graph's general shape and direction.**Note:** The zeros of the polynomial are critical points where the graph intersects the x-axis. Understanding these can help in sketching an accurate graph.

Answered: Find the 2eros and graph 4=-x5-x年ay

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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**Task: Finding the Zeros and Graphing a Polynomial**

Given the polynomial equation:

\[ y = -x^5 - x^4 + 2x^3 + 2x^2 - x - 1 \]

**Objective:**

1. **Find the Zeros:**
- Determine the values of $ x $ for which $ y = 0 $.
- Use techniques such as synthetic division, factoring, or the Rational Root Theorem where applicable.

2. **Graph the Polynomial:**
- Plot the function on a coordinate plane.
- Identify key features such as intercepts, turning points, and end behavior.
- Analyze the degree and leading coefficient to determine the graph's general shape and direction.

**Note:** The zeros of the polynomial are critical points where the graph intersects the x-axis. Understanding these can help in sketching an accurate graph.

Expert Solution

Step 1

Given equation,

$y = - x^{5} - x^{4} + 2 x^{3} + 2 x^{2} - x - 1 T o f i n d z e r o s o f e q u a t i o n, e q u a t e i t t o 0 - x^{5} - x^{4} + 2 x^{3} + 2 x^{2} - x - 1 = 0 - x^{4} (x + 1) + 2 x^{2} (x + 1) - 1 (x + 1) = 0 (x + 1) (- x^{4} + 2 x^{2} - 1) = 0 (x + 1) (x^{4} - 2 x^{2} + 1) = 0 (x + 1) {(x^{2} - 1)}^{2} = 0 (x + 1) {((x - 1) (x + 1))}^{2} = 0 (x + 1) {(x - 1)}^{2} {(x + 1)}^{2} = 0 {(x - 1)}^{2} {(x + 1)}^{3} = 0$

Roots are 1, 1, -1, -1 & -1.

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