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:
\[ f(-2) = -3 \]
