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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![**Problem Statement:**
Find a polynomial function with real coefficients that has the given zeros. (There are many correct answers.)
**Given Zeros:**
\[ \frac{3}{8}, -1, 9, \sqrt{3} \]
**Function Representation:**
\[ f(x) = \boxed{} \]
**Explanation for Students:**
To find the polynomial function with the given zeros, you will need to create factors of the form \((x - \text{zero})\) for each zero. For example, if one of the zeros is \( \frac{3}{8} \), the factor would be \((x - \frac{3}{8})\).
- The zeros \( \frac{3}{8}, -1, 9, \text{ and } \sqrt{3} \) lead to the factors:
- \((x - \frac{3}{8})\)
- \((x + 1)\)
- \((x - 9)\)
- \((x - \sqrt{3})\)
- Notice that since all the zeros given are real, they directly become linear factors of the polynomial.
Multiply these factors to find the polynomial:
\[ f(x) = (x - \frac{3}{8})(x + 1)(x - 9)(x - \sqrt{3}) \]
You can expand this product to find an explicit polynomial expression.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a744944-d6d1-44b6-81cf-23b766613f36%2Feb43369d-3082-46b9-9db5-b98632a18b2a%2F5g44vri_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find a polynomial function with real coefficients that has the given zeros. (There are many correct answers.)
**Given Zeros:**
\[ \frac{3}{8}, -1, 9, \sqrt{3} \]
**Function Representation:**
\[ f(x) = \boxed{} \]
**Explanation for Students:**
To find the polynomial function with the given zeros, you will need to create factors of the form \((x - \text{zero})\) for each zero. For example, if one of the zeros is \( \frac{3}{8} \), the factor would be \((x - \frac{3}{8})\).
- The zeros \( \frac{3}{8}, -1, 9, \text{ and } \sqrt{3} \) lead to the factors:
- \((x - \frac{3}{8})\)
- \((x + 1)\)
- \((x - 9)\)
- \((x - \sqrt{3})\)
- Notice that since all the zeros given are real, they directly become linear factors of the polynomial.
Multiply these factors to find the polynomial:
\[ f(x) = (x - \frac{3}{8})(x + 1)(x - 9)(x - \sqrt{3}) \]
You can expand this product to find an explicit polynomial expression.
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