Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
Write the polynomial in factored form as a product of linear factors.

Transcribed Image Text:**Problem Statement:**
Find all real zeros of the polynomial function \( P(x) = 3x^3 - 5x^2 - 8x - 2 \).
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**Explanation:**
To find the real zeros of a polynomial, you are looking for the values of \( x \) that satisfy the equation \( P(x) = 0 \). The zeros of the polynomial can be found using various methods such as:
1. **Graphical Method:** Plotting the function and identifying the x-intercepts.
2. **Analytical Methods:**
- Factoring (if possible)
- Using the Rational Root Theorem
- Synthetic Division
- Numerical methods, such as the Newton-Raphson method for more complex polynomials
**Steps to Solve:**
1. **Graph the Function:**
- Determine where the curve intersects the x-axis.
- Use graphing tools or technology to approximate the points of intersection.
2. **Test Possible Rational Roots:**
- According to the Rational Root Theorem, test potential rational roots.
- Use synthetic division to confirm any viable candidates.
3. **Derive Exact Roots:**
- If rational roots are found, factor them out.
- Solve the remaining simplified polynomial.
By finding the roots of the polynomial, you determine the values of \( x \) that make the polynomial equal to zero. These roots may include rational numbers, irrational numbers, or complex pairs.
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This explanation gives an overview of how to approach finding real zeros for the given polynomial function on an educational platform.
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