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State how many complex and how many real zeros the function has. f(x) = x² + 6x² - 4x² + 72x-192

State how many complex and how many real zeros the function has. f(x) = x² + 6x² - 4x² + 72x-192

Calculus: Early Transcendentals
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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State how many complex and how many real zeros the function has

f(x)= x4 + 6x3 - 4x2 + 72x - 192

**Question:** State how many complex and how many real zeros the function has. \( f(x) = x^4 + 6x^3 - 4x^2 + 72x - 192 \) **Explanation:** The question asks for the determination of complex and real zeros in a polynomial function. The function given is a quartic polynomial, \( f(x) = x^4 + 6x^3 - 4x^2 + 72x - 192 \). To solve this, one would typically use the following steps: 1. **Determine the Degree:** The leading term \( x^4 \) indicates that the polynomial is of degree 4, suggesting there are 4 zeros in total. 2. **Factorization and Solutions:** - Find any rational zeros using the Rational Root Theorem. - Use polynomial division to simplify the polynomial. - Solve for the remaining polynomial to find all real and complex roots. 3. **Calculate Real and Complex Zeros:** - Zeros can be real (which can be further classified into rational or irrational) or complex (not real). The complex zeros occur in conjugate pairs. This serves as a practical application of complex number and polynomial function theory in algebra.
Transcribed Image Text:**Question:** State how many complex and how many real zeros the function has. \( f(x) = x^4 + 6x^3 - 4x^2 + 72x - 192 \) **Explanation:** The question asks for the determination of complex and real zeros in a polynomial function. The function given is a quartic polynomial, \( f(x) = x^4 + 6x^3 - 4x^2 + 72x - 192 \). To solve this, one would typically use the following steps: 1. **Determine the Degree:** The leading term \( x^4 \) indicates that the polynomial is of degree 4, suggesting there are 4 zeros in total. 2. **Factorization and Solutions:** - Find any rational zeros using the Rational Root Theorem. - Use polynomial division to simplify the polynomial. - Solve for the remaining polynomial to find all real and complex roots. 3. **Calculate Real and Complex Zeros:** - Zeros can be real (which can be further classified into rational or irrational) or complex (not real). The complex zeros occur in conjugate pairs. This serves as a practical application of complex number and polynomial function theory in algebra.
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