Find the polynomial function (in simplest standard form) that would give a graph like the one shown. It crosses the x-axis at (-3, 0), (2, 0), and (5, 0). ful x
Find the polynomial function (in simplest standard form) that would give a graph like the one shown. It crosses the x-axis at (-3, 0), (2, 0), and (5, 0). ful x
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,...
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![### Polynomial Function from a Graph
**Problem Statement:**
Find the polynomial function (in simplest standard form) that would give a graph like the one shown. It crosses the x-axis at (-3, 0), (2, 0), and (5, 0).
**Graph Description:**
The provided graph illustrates a polynomial curve that intersects the x-axis at three points: (-3, 0), (2, 0), and (5, 0). The y-axis is labeled with 'y' and the x-axis with 'x'. The polynomial appears to have a turning point or local extremum between the intercepts.
**Steps to Solution:**
Given the x-intercepts, the polynomial can be written in factored form as:
\[ P(x) = a(x + 3)(x - 2)(x - 5) \]
Here, \( a \) is a constant that can be any real number. To determine \( a \), additional information such as a point on the curve other than the intercepts would be required. In this case, if no such additional point is given, we can assume \( a = 1 \) for simplicity:
\[ P(x) = (x + 3)(x - 2)(x - 5) \]
To find the simplest standard form, expand the factors:
1. Multiply \( (x + 3) \) and \( (x - 2) \):
\[ (x + 3)(x - 2) = x^2 - 2x + 3x - 6 = x^2 + x - 6 \]
2. Multiply the result by \( (x - 5) \):
\[ (x^2 + x - 6)(x - 5) = x^3 - 5x^2 + x^2 - 5x - 6x + 30 \]
3. Combine like terms:
\[ x^3 - 4x^2 - 11x + 30 \]
Thus, the polynomial in simplest standard form is:
\[ P(x) = x^3 - 4x^2 - 11x + 30 \]
**Final Polynomial:**
\[ P(x) = x^3 - 4x^2 - 11x + 30 \]
This polynomial function satisfies the given x-intercepts as shown in the graph.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3692e55-09d2-4c7f-90e9-0ed90d8b0dce%2F8703880c-047e-40f1-beec-1fe6af4b6de0%2Fh3ixg6v_processed.png&w=3840&q=75)
Transcribed Image Text:### Polynomial Function from a Graph
**Problem Statement:**
Find the polynomial function (in simplest standard form) that would give a graph like the one shown. It crosses the x-axis at (-3, 0), (2, 0), and (5, 0).
**Graph Description:**
The provided graph illustrates a polynomial curve that intersects the x-axis at three points: (-3, 0), (2, 0), and (5, 0). The y-axis is labeled with 'y' and the x-axis with 'x'. The polynomial appears to have a turning point or local extremum between the intercepts.
**Steps to Solution:**
Given the x-intercepts, the polynomial can be written in factored form as:
\[ P(x) = a(x + 3)(x - 2)(x - 5) \]
Here, \( a \) is a constant that can be any real number. To determine \( a \), additional information such as a point on the curve other than the intercepts would be required. In this case, if no such additional point is given, we can assume \( a = 1 \) for simplicity:
\[ P(x) = (x + 3)(x - 2)(x - 5) \]
To find the simplest standard form, expand the factors:
1. Multiply \( (x + 3) \) and \( (x - 2) \):
\[ (x + 3)(x - 2) = x^2 - 2x + 3x - 6 = x^2 + x - 6 \]
2. Multiply the result by \( (x - 5) \):
\[ (x^2 + x - 6)(x - 5) = x^3 - 5x^2 + x^2 - 5x - 6x + 30 \]
3. Combine like terms:
\[ x^3 - 4x^2 - 11x + 30 \]
Thus, the polynomial in simplest standard form is:
\[ P(x) = x^3 - 4x^2 - 11x + 30 \]
**Final Polynomial:**
\[ P(x) = x^3 - 4x^2 - 11x + 30 \]
This polynomial function satisfies the given x-intercepts as shown in the graph.
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