Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Question

Write the polynomial in factored form as a product of linear factors.

**Problem Statement:** Find all real zeros of the polynomial function \( P(x) = 3x^3 - 5x^2 - 8x - 2 \). --- **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. --- 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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Transcribed Image Text:**Problem Statement:** Find all real zeros of the polynomial function \( P(x) = 3x^3 - 5x^2 - 8x - 2 \). --- **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. --- 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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