The function below has at least one rational zero. Use this fact to find all zeros of the function. 2 g(x) = 5x³-8x² - 2x+3 If there is more than one zero, separate them with commas. Write exact values, not decimal approximations.
The function below has at least one rational zero. Use this fact to find all zeros of the function. 2 g(x) = 5x³-8x² - 2x+3 If there is more than one zero, separate them with commas. Write exact values, not decimal approximations.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Finding Zeros of a Polynomial Function
The function below has at least one rational zero. Use this fact to find all zeros of the function.
\[ g(x) = 5x^3 - 8x^2 - 2x + 3 \]
**Instructions:**
If there is more than one zero, separate them with commas. Write exact values, not decimal approximations.
**Input Box:**
Below the instructions, there is an input box provided for you to enter the zeros of the function. There are also a few mathematical symbol buttons next to the input box to assist you in entering your answers precisely, including symbols for addition, subtraction, multiplication, division, exponents, and roots.
---
**Explanation of Rational Zero Theorem (Optional Information for Better Understanding):**
To find the rational zeros of the polynomial, you can apply the Rational Zero Theorem, which states that any rational zero of the polynomial \( g(x) = 5x^3 - 8x^2 - 2x + 3 \) must be a fraction \(\frac{p}{q}\), where \( p \) is a factor of the constant term (3), and \( q \) is a factor of the leading coefficient (5).
The factors of the constant term \(3\) are: \(\pm 1, \pm 3 \).
The factors of the leading coefficient \(5\) are: \(\pm 1, \pm 5 \).
Therefore, possible rational zeros are: \(\pm 1, \pm 3, \pm \frac{1}{5}, \pm \frac{3}{5} \).
By substituting these values into the polynomial, you can determine which, if any, are actual zeros.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F462ffa92-ffdf-4e48-9e22-7b7cea535b4f%2F890e3b98-cd38-48f3-95ae-6bfa7b924439%2Fgq17dfj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding Zeros of a Polynomial Function
The function below has at least one rational zero. Use this fact to find all zeros of the function.
\[ g(x) = 5x^3 - 8x^2 - 2x + 3 \]
**Instructions:**
If there is more than one zero, separate them with commas. Write exact values, not decimal approximations.
**Input Box:**
Below the instructions, there is an input box provided for you to enter the zeros of the function. There are also a few mathematical symbol buttons next to the input box to assist you in entering your answers precisely, including symbols for addition, subtraction, multiplication, division, exponents, and roots.
---
**Explanation of Rational Zero Theorem (Optional Information for Better Understanding):**
To find the rational zeros of the polynomial, you can apply the Rational Zero Theorem, which states that any rational zero of the polynomial \( g(x) = 5x^3 - 8x^2 - 2x + 3 \) must be a fraction \(\frac{p}{q}\), where \( p \) is a factor of the constant term (3), and \( q \) is a factor of the leading coefficient (5).
The factors of the constant term \(3\) are: \(\pm 1, \pm 3 \).
The factors of the leading coefficient \(5\) are: \(\pm 1, \pm 5 \).
Therefore, possible rational zeros are: \(\pm 1, \pm 3, \pm \frac{1}{5}, \pm \frac{3}{5} \).
By substituting these values into the polynomial, you can determine which, if any, are actual zeros.
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