< The polynomial function is f(x) = (Simplify your answer.) Question 17, 3.3.59 > Find a polynomial function f(x) of degree 3 with real coefficients that satisfies the following conditions Zero of 0 and zero of 4 having multiplicity 2; f(5) = 25 ...
< The polynomial function is f(x) = (Simplify your answer.) Question 17, 3.3.59 > Find a polynomial function f(x) of degree 3 with real coefficients that satisfies the following conditions Zero of 0 and zero of 4 having multiplicity 2; f(5) = 25 ...
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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![### Question 17, 3.3.59
**Problem Statement:**
Find a polynomial function \( f(x) \) of degree 3 with real coefficients that satisfies the following conditions:
- Zero of 0 and zero of 4 having multiplicity 2; \( f(5) = 25 \)
**Solution Approach:**
To find the polynomial function, consider the given zeros and their multiplicities:
1. Zero at \( x = 0 \)
2. Zero at \( x = 4 \) with multiplicity 2
This implies the polynomial can be expressed as:
\[ f(x) = a(x)(x-4)^2 \]
Given that:
\[ f(5) = 25 \]
Use this information to determine the value of \( a \). Substitute \( x = 5 \) into the polynomial and solve for \( a \):
\[ f(5) = a(5)(5-4)^2 = a(5)(1) = 5a \]
\[ 5a = 25 \]
\[ a = 5 \]
Thus, the polynomial function is:
\[ f(x) = 5x(x-4)^2 \]
**Final Answer:**
The polynomial function is \( f(x) = 5x(x-4)^2 \). (Simplify your answer.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F641fd7f9-b649-4930-9241-4a5557bbdfb1%2F1ba0b768-36be-46c8-88a6-58ba980a67e9%2Frmx0snw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Question 17, 3.3.59
**Problem Statement:**
Find a polynomial function \( f(x) \) of degree 3 with real coefficients that satisfies the following conditions:
- Zero of 0 and zero of 4 having multiplicity 2; \( f(5) = 25 \)
**Solution Approach:**
To find the polynomial function, consider the given zeros and their multiplicities:
1. Zero at \( x = 0 \)
2. Zero at \( x = 4 \) with multiplicity 2
This implies the polynomial can be expressed as:
\[ f(x) = a(x)(x-4)^2 \]
Given that:
\[ f(5) = 25 \]
Use this information to determine the value of \( a \). Substitute \( x = 5 \) into the polynomial and solve for \( a \):
\[ f(5) = a(5)(5-4)^2 = a(5)(1) = 5a \]
\[ 5a = 25 \]
\[ a = 5 \]
Thus, the polynomial function is:
\[ f(x) = 5x(x-4)^2 \]
**Final Answer:**
The polynomial function is \( f(x) = 5x(x-4)^2 \). (Simplify your answer.)
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