let f(x) be a polynomial satisfying f(x) : f(1/x) = f(x) + f(1/x) for all `XinR` :- {O} and f(5) =126, then find f(3).

Answer

To solve the problem step by step, we will analyze the given functional equation and use the information provided to find the polynomial \( f(x) \).

### Step 1: Understand the functional equation
We are given:
\[ f(x) \cdot f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right) \]
for all \( x \in \mathbb{R} \setminus \{0\} \).

### Step 2: Substitute \( x \) with \( \frac{1}{x} \)
Substituting \( x \) with \( \frac{1}{x} \) in the functional equation gives:
\[ f\left(\frac{1}{x}\right) \cdot f(x) = f\left(\frac{1}{x}\right) + f(x) \]
This equation is identical to the original equation, confirming that the relationship holds for both \( f(x) \) and \( f\left(\frac{1}{x}\right) \).

### Step 3: Rearranging the equation
From the original equation, we can rearrange it to isolate \( f(x) \):
\[ f(x) \cdot f\left(\frac{1}{x}\right) - f(x) - f\left(\frac{1}{x}\right) = 0 \]
This can be factored as:
\[ (f(x) - 1)(f\left(\frac{1}{x}\right) - 1) = 1 \]

### Step 4: Define a new function
Let \( g(x) = f(x) - 1 \). Then the equation becomes:
\[ g(x) \cdot g\left(\frac{1}{x}\right) = 1 \]
This implies that \( g(x) \) and \( g\left(\frac{1}{x}\right) \) are multiplicative inverses.

### Step 5: Determine the form of \( g(x) \)
Since \( g(x) \) and \( g\left(\frac{1}{x}\right) \) are inverses, we can assume \( g(x) = x^n \) for some integer \( n \). This gives us:
\[ g\left(\frac{1}{x}\right) = \left(\frac{1}{x}\right)^n = \frac{1}{x^n} \]
Thus, we have:
\[ x^n \cdot \frac{1}{x^n} = 1 \]
which holds true.

### Step 6: Relate back to \( f(x) \)
Since \( g(x) = x^n \), we have:
\[ f(x) = g(x) + 1 = x^n + 1 \]

### Step 7: Use the given condition \( f(5) = 126 \)
We know:
\[ f(5) = 5^n + 1 = 126 \]
This simplifies to:
\[ 5^n = 125 \]
Thus, \( n = 3 \).

### Step 8: Write the polynomial function
Now we can write:
\[ f(x) = x^3 + 1 \]

### Step 9: Find \( f(3) \)
To find \( f(3) \):
\[ f(3) = 3^3 + 1 = 27 + 1 = 28 \]

### Final Answer
Thus, the value of \( f(3) \) is:
\[ \boxed{28} \]