12) List the potential rational zeros of the polynomial function. Do not find the zeros. f (x) = -3x2 – 2x + 8
12) List the potential rational zeros of the polynomial function. Do not find the zeros. f (x) = -3x2 – 2x + 8
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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![### Problem 12: Potential Rational Zeros of a Polynomial Function
**Instruction:**
List the potential rational zeros of the polynomial function. Do not find the zeros.
**Polynomial Function:**
\[ f(x) = -3x^2 - 2x + 8 \]
To list the potential rational zeros for the polynomial \( f(x) = -3x^2 - 2x + 8 \), we use the Rational Root Theorem. This theorem states that any potential rational zero, expressed as a fraction \(\frac{p}{q}\), is such that \(p\) is a factor of the constant term (8 in this case), and \(q\) is a factor of the leading coefficient (-3 in this case).
**Factors of 8 (constant term):** \( \pm 1, \pm 2, \pm 4, \pm 8 \)
**Factors of -3 (leading coefficient):** \( \pm 1, \pm 3 \)
**Potential Rational Zeros:**
The potential rational zeros are all possible fractions \(\frac{p}{q}\), where \(p\) is a factor of 8, and \(q\) is a factor of -3. These include:
\[
\pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3}, \pm \frac{8}{3}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F89d7554b-e759-4ec2-8c2a-403e3f2f0927%2F117e9d0e-4f13-46f5-8e5d-c84c7937a0e5%2F2woj7d_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 12: Potential Rational Zeros of a Polynomial Function
**Instruction:**
List the potential rational zeros of the polynomial function. Do not find the zeros.
**Polynomial Function:**
\[ f(x) = -3x^2 - 2x + 8 \]
To list the potential rational zeros for the polynomial \( f(x) = -3x^2 - 2x + 8 \), we use the Rational Root Theorem. This theorem states that any potential rational zero, expressed as a fraction \(\frac{p}{q}\), is such that \(p\) is a factor of the constant term (8 in this case), and \(q\) is a factor of the leading coefficient (-3 in this case).
**Factors of 8 (constant term):** \( \pm 1, \pm 2, \pm 4, \pm 8 \)
**Factors of -3 (leading coefficient):** \( \pm 1, \pm 3 \)
**Potential Rational Zeros:**
The potential rational zeros are all possible fractions \(\frac{p}{q}\), where \(p\) is a factor of 8, and \(q\) is a factor of -3. These include:
\[
\pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3}, \pm \frac{8}{3}
\]
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