Let f(x) be a polynomial. When f(x) is divided by(x-2) , the remainder is 5. If f(x) is divisible by (x+3), find the remainder when f(x) is divisible by (x-2)(x+3)

Given the polynomial \( f(x) \), we know the following:

1. When \( f(x) \) is divided by \( (x-2) \), the remainder is 5. This implies \( f(2) = 5 \).
2. \( f(x) \) is divisible by \( (x+3) \), which implies \( f(-3) = 0 \).

We need to find the remainder when \( f(x) \) is divided by \( (x-2)(x+3) \).

### Step 1: Expressing the Remainder
When dividing \( f(x) \) by \( (x-2)(x+3) \), the remainder must be a linear polynomial, say \( R(x) = ax + b \). Hence, we can write:
\[
f(x) = (x-2)(x+3)q(x) + ax + b
\]
for some quotient polynomial \( q(x) \).

### Step 2: Use Given Conditions
Using the conditions given:

- \( f(2) = 5 \), substitute \( x = 2 \) into the equation:
  \[
  f(2) = (2-2)(2+3)q(2) + 2a + b = 5
  \]
  Simplifying, we get:
  \[
  2a + b = 5 \quad \text{(Equation 1)}
  \]

- \( f(-3) = 0 \), substitute \( x = -3 \) into the equation:
  \[
  f(-3) = (-3-2)(-3+3)q(-3) + (-3a + b) = 0
  \]
  Simplifying, we get:
  \[
  -3a + b = 0 \quad \text{(Equation 2)}
  \]

### Step 3: Solve the System of Equations
Now, solve the system of linear equations:
\[
\begin{aligned}
2a + b & = 5 \quad \text{(Equation 1)} \\
-3a + b & = 0 \quad \text{(Equation 2)}
\end{aligned}
\]

Subtract Equation 2 from Equation 1:
\[
(2a + b) - (-3a + b) = 5 - 0
\]
\[
5a = 5
\]
\[
a = 1
\]

Now, substitute \( a = 1 \) into Equation 2:
\[
-3(1) + b = 0
\]
\[
b = 3
\]

### Step 4: Write the Remainder
The remainder \( R(x) \) when \( f(x) \) is divided by \( (x-2)(x+3) \) is:
\[
R(x) = ax + b = x + 3
\]

### Final Answer
The remainder when \( f(x) \) is divided by \( (x-2)(x+3) \) is \( \boxed{x + 3} \).

Would you like more details or have any questions? Here are some related questions to explore:

1. How can you determine the quotient \( q(x) \) in the division of \( f(x) \)?
2. What is the Remainder Theorem and how does it apply to polynomial division?
3. What happens if the divisor is a quadratic polynomial instead of linear?
4. How do the conditions \( f(2) = 5 \) and \( f(-3) = 0 \) help in determining the remainder?
5. Can the remainder be a quadratic polynomial? Why or why not?
6. How does synthetic division relate to this problem?
7. What would be the remainder if \( f(x) \) were divided by \( (x-2)(x+3)(x-1) \)?
8. What is the significance of roots and factors in polynomial division?

**Tip:** When solving polynomial division problems, always verify if additional conditions (like roots or specific function values) are provided, as they simplify finding the remainder or quotient.

Explore how to find the remainder when a polynomial is divided by (x-2)(x+3) given specific conditions. This problem involves polynomial division, the remainder theorem, and solving linear equations.

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