Let g(x) be a polynomial function satisfying g(x).g(y) = g(x) + g(y) + g(xy) -2 for all x, `y in R` and `g(1) != 1`. If g(3) = 10 then g(5) equals

Options

A. `-24`
B. 16
C. 26
D. 34

Answer

To solve the problem, we need to find the polynomial function \( g(x) \) that satisfies the given functional equation and then use it to find \( g(5) \).

### Step-by-Step Solution:

1. **Understanding the Functional Equation:**
We are given that:
\[ g(x) g(y) = g(x) + g(y) + g(xy) - 2 \]
for all \( x, y \in \mathbb{R} \).

2. **Substituting Values:**
Let's substitute \( x = 3 \) and \( y = 1 \) into the equation. We know \( g(3) = 10 \).
\[ g(3) g(1) = g(3) + g(1) + g(3 \cdot 1) - 2 \]
This simplifies to:
\[ 10 g(1) = 10 + g(1) + 10 - 2 \]
\[ 10 g(1) = 18 + g(1) \]

3. **Rearranging the Equation:**
Rearranging gives:
\[ 10 g(1) - g(1) = 18 \]
\[ 9 g(1) = 18 \]
Dividing both sides by 9:
\[ g(1) = 2 \]

4. **Finding a Pattern:**
We notice that \( g(1) = 2 \) and \( g(3) = 10 \). Let's check if there is a polynomial pattern. We hypothesize that \( g(x) \) might be of the form \( g(x) = x^2 + 1 \).

5. **Verifying the Hypothesis:**
If \( g(x) = x^2 + 1 \), then:
- \( g(1) = 1^2 + 1 = 2 \) (which is correct),
- \( g(3) = 3^2 + 1 = 9 + 1 = 10 \) (which is also correct).

6. **Finding \( g(5) \):**
Now, we can find \( g(5) \):
\[ g(5) = 5^2 + 1 = 25 + 1 = 26 \]

7. **Conclusion:**
Therefore, the value of \( g(5) \) is:
\[ \boxed{26} \]