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
The graph of the polynomial f(x) is given below. If f(x) has degree 3, find the factored equation for f(x).
![### Educational Content: Polynomial Functions
#### Question
The graph of the polynomial \( f(x) \) is given below. If \( f(x) \) has degree 3, find the factored equation for \( f(x) \).
#### Graph Description
The graph displayed is a curve representing the polynomial function \( f(x) \). Key features of the graph include:
- **Intercepts**: The graph crosses the x-axis at three points, approximately near \( x = 0 \), \( x = 2 \), and \( x = 5 \). These points indicate potential roots or solutions for the polynomial.
- **Behavior**: The graph ascends sharply from the left, passes through the x-axis, levels out slightly, and then descends again, crossing the x-axis. This pattern is characteristic of a cubic polynomial.
- **Coordinates**: The y-axis ranges from -1 to 6, while the x-axis ranges from -2 to 7.
#### Explanation
Given that \( f(x) \) is a degree 3 polynomial, it can be expressed in the factored form:
\[ f(x) = a(x-r_1)(x-r_2)(x-r_3) \]
where \( r_1 \), \( r_2 \), and \( r_3 \) are the roots of the polynomial. Based on the graph, the roots appear to be at \( x = 0 \), \( x = 2 \), and \( x = 5 \).
The challenge is to determine the value of \( a \), which affects the vertical stretch of the graph, by examining another known point on the curve or additional conditions.
##### Note
This task involves analyzing and interpreting a graphical representation of a polynomial function to deduce its algebraic form, emphasizing connections between algebraic and graphical representations in the study of mathematics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff0dbddf9-08e8-446e-97c9-78b49c293188%2F633f8213-86b8-4f33-800d-f1ad5a7283f1%2F5rjio3n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Content: Polynomial Functions
#### Question
The graph of the polynomial \( f(x) \) is given below. If \( f(x) \) has degree 3, find the factored equation for \( f(x) \).
#### Graph Description
The graph displayed is a curve representing the polynomial function \( f(x) \). Key features of the graph include:
- **Intercepts**: The graph crosses the x-axis at three points, approximately near \( x = 0 \), \( x = 2 \), and \( x = 5 \). These points indicate potential roots or solutions for the polynomial.
- **Behavior**: The graph ascends sharply from the left, passes through the x-axis, levels out slightly, and then descends again, crossing the x-axis. This pattern is characteristic of a cubic polynomial.
- **Coordinates**: The y-axis ranges from -1 to 6, while the x-axis ranges from -2 to 7.
#### Explanation
Given that \( f(x) \) is a degree 3 polynomial, it can be expressed in the factored form:
\[ f(x) = a(x-r_1)(x-r_2)(x-r_3) \]
where \( r_1 \), \( r_2 \), and \( r_3 \) are the roots of the polynomial. Based on the graph, the roots appear to be at \( x = 0 \), \( x = 2 \), and \( x = 5 \).
The challenge is to determine the value of \( a \), which affects the vertical stretch of the graph, by examining another known point on the curve or additional conditions.
##### Note
This task involves analyzing and interpreting a graphical representation of a polynomial function to deduce its algebraic form, emphasizing connections between algebraic and graphical representations in the study of mathematics.
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