Polynomial `P(x)` contains only terms of aodd degree. when `P(x)` is divided by `(x - 3)`, the ramainder is `6`. If `P(x)` is divided by `(x^(2) - 9)` then remainder is `g(x)`. Then find the value of `g(2)`.

Explanation

To solve the problem, we need to find the value of \( g(2) \) where \( g(x) \) is the remainder when the polynomial \( P(x) \) is divided by \( x^2 - 9 \).

### Step-by-Step Solution:

1. **Understand the Polynomial**:
Since \( P(x) \) contains only terms of odd degree, it can be expressed in the general form of odd degree polynomials, such as \( P(x) = a_n x^n + a_{n-2} x^{n-2} + \ldots + a_1 x \).

2. **Remainder Theorem**:
According to the Remainder Theorem, when \( P(x) \) is divided by \( (x - 3) \), the remainder is \( P(3) \). We know that \( P(3) = 6 \).

3. **Dividing by \( x^2 - 9 \)**:
The polynomial \( x^2 - 9 \) can be factored as \( (x - 3)(x + 3) \). When dividing \( P(x) \) by \( x^2 - 9 \), the remainder \( g(x) \) must be a polynomial of degree less than 2. Therefore, we can express \( g(x) \) in the form:
\[ g(x) = ax + b \]

4. **Finding \( g(3) \)**:
Since \( P(x) = (x^2 - 9)Q(x) + g(x) \), substituting \( x = 3 \):
\[ P(3) = g(3) \]
We already know \( P(3) = 6 \), so:
\[ g(3) = 6 \]

5. **Finding \( g(-3) \)**:
Similarly, substituting \( x = -3 \):
\[ P(-3) = g(-3) \]
Since \( P(x) \) is an odd function (only contains odd degree terms), we have:
\[ P(-3) = -P(3) = -6 \]
Thus:
\[ g(-3) = -6 \]

6. **Setting Up the System of Equations**:
Now we have two equations:
- From \( g(3) = 6 \):
\[ 3a + b = 6 \quad \text{(1)} \]
- From \( g(-3) = -6 \):
\[ -3a + b = -6 \quad \text{(2)} \]

7. **Solving the System of Equations**:
Subtract equation (2) from equation (1):
\[ (3a + b) - (-3a + b) = 6 - (-6) \]
This simplifies to:
\[ 6a = 12 \implies a = 2 \]
Now substitute \( a = 2 \) back into equation (1):
\[ 3(2) + b = 6 \implies 6 + b = 6 \implies b = 0 \]

8. **Finding \( g(x) \)**:
Thus, we have:
\[ g(x) = 2x + 0 = 2x \]

9. **Calculating \( g(2) \)**:
Finally, we need to find \( g(2) \):
\[ g(2) = 2(2) = 4 \]

### Final Answer:
\[ g(2) = 4 \]