Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
![**Finding Rational Roots of Polynomials**
The possible rational roots of
\[ f(x) = 5x^3 - 8x^2 - 4x + 3 \]
are \( x = \) [Input Box]
(Use "+-" to indicate the ± symbol. So "+-1" becomes \( \pm 1 \).)
---
In this exercise, you are tasked to find and indicate the possible rational roots of the polynomial function \( f(x) = 5x^3 - 8x^2 - 4x + 3 \).
### Step-by-Step Instructions:
1. **Identify the Polynomial and Coefficients:**
- Observe and note down the polynomial \( f(x) \) given.
- Identify the coefficients of each term in the polynomial.
2. **Utilize the Rational Root Theorem:**
- According to the Rational Root Theorem, the possible rational roots of a polynomial are given by:
\[
\text{Possible Rational Roots} = \pm \left( \frac{\text{Factors of the Constant Term}}{\text{Factors of the Leading Coefficient}} \right)
\]
- In this polynomial:
- The constant term is 3, so the factors are \( \pm 1, \pm 3 \).
- The leading coefficient is 5, so the factors are \( \pm 1, \pm 5 \).
3. **List Possible Rational Roots:**
- From the factors identified, form all possible rational roots.
- These roots are in the form of \( \pm \left( \frac{\text{Factors of 3}}{\text{Factors of 5}} \right) \):
\[
\pm 1, \pm \frac{1}{5}, \pm 3, \pm \frac{3}{5}
\]
### Example Input and Output:
- For example, if asked to input ±1 as possible rational roots, you would write:
\[
x = \pm 1
\]
Explore and verify your solutions using the polynomial \( f(x) = 5x^3 - 8x^2 - 4x + 3 \) by substituting the roots back into the polynomial to see if it equals zero, thus confirming the rational roots.
### Practice:
Using the input box, list down all possible rational roots](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcae97c47-da87-4dbe-a1b5-55efe44ed72d%2F4e0c62b3-3f59-4f5d-9194-625df1fc22ec%2Fwlrd81l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding Rational Roots of Polynomials**
The possible rational roots of
\[ f(x) = 5x^3 - 8x^2 - 4x + 3 \]
are \( x = \) [Input Box]
(Use "+-" to indicate the ± symbol. So "+-1" becomes \( \pm 1 \).)
---
In this exercise, you are tasked to find and indicate the possible rational roots of the polynomial function \( f(x) = 5x^3 - 8x^2 - 4x + 3 \).
### Step-by-Step Instructions:
1. **Identify the Polynomial and Coefficients:**
- Observe and note down the polynomial \( f(x) \) given.
- Identify the coefficients of each term in the polynomial.
2. **Utilize the Rational Root Theorem:**
- According to the Rational Root Theorem, the possible rational roots of a polynomial are given by:
\[
\text{Possible Rational Roots} = \pm \left( \frac{\text{Factors of the Constant Term}}{\text{Factors of the Leading Coefficient}} \right)
\]
- In this polynomial:
- The constant term is 3, so the factors are \( \pm 1, \pm 3 \).
- The leading coefficient is 5, so the factors are \( \pm 1, \pm 5 \).
3. **List Possible Rational Roots:**
- From the factors identified, form all possible rational roots.
- These roots are in the form of \( \pm \left( \frac{\text{Factors of 3}}{\text{Factors of 5}} \right) \):
\[
\pm 1, \pm \frac{1}{5}, \pm 3, \pm \frac{3}{5}
\]
### Example Input and Output:
- For example, if asked to input ±1 as possible rational roots, you would write:
\[
x = \pm 1
\]
Explore and verify your solutions using the polynomial \( f(x) = 5x^3 - 8x^2 - 4x + 3 \) by substituting the roots back into the polynomial to see if it equals zero, thus confirming the rational roots.
### Practice:
Using the input box, list down all possible rational roots
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