homework53
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School
Georgia Institute Of Technology *
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Course
1S
Subject
Mathematics
Date
Feb 20, 2024
Type
docx
Pages
2
Uploaded by ElderMantis2561
Certainly! Here's another math problem:
Problem:
Let \( f(x) \) be a polynomial function such that when \( f(x) \) is divided by \( x - 3 \), the remainder is 6, and when \( f(x) \) is divided by \( x + 2 \), the remainder is -3. Find the remainder when \( f(x) \) is divided by \( (x - 3)(x + 2) \).
Solution:
Let \( f(x) \) be the polynomial function of degree \( n \).
According to the Remainder Theorem, when \( f(x) \) is divided by \( x - 3 \), the remainder is \( f(3) = 6 \).
Similarly, when \( f(x) \) is divided by \( x + 2 \), the remainder is \( f(-2) = -3 \).
Using these pieces of information, we can set up a system of equations:
1. \( f(3) = 6 \)
2. \( f(-2) = -3 \)
Now, let's construct a polynomial function \( f(x) \) that satisfies these conditions.
From the first equation:
\[ f(3) = 6 \]
\[ \Rightarrow a(3)^n + b(3)^{n-1} + \ldots + c = 6 \]
From the second equation:
\[ f(-2) = -3 \]
\[ \Rightarrow a(-2)^n + b(-2)^{n-1} + \ldots + c = -3 \]
Now, we need to solve this system of equations to find the coefficients \( a, b, c, \ldots \) of the polynomial function \( f(x) \).
Once we find the polynomial function \( f(x) \), we can easily find the remainder when \( f(x) \) is divided
by \( (x - 3)(x + 2) \) by substituting \( x = 3 \) or \( x = -2 \) into \( f(x) \). The remainder will be the value of \( f(x) \) at that point.
Let me know if you need assistance solving the system of equations or if you have any questions about the process!
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