A polynomial function f(x) satisfies the condition `f(x)f(1/x)=f(x)+f(1/x)` for all `x inR`,`x!=0`. If f(3)=-26, then f(4)=

Options

A. `-35`
B. `-63`
C. 65
D. none of these

Explanation

To solve the problem step by step, we start with the given condition for the polynomial function \( f(x) \):

### Step 1: Understand the condition
The condition given is:
\[ f(x)f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right) \]
for all \( x \in \mathbb{R} \) and \( x \neq 0 \).

### Step 2: Assume a form for \( f(x) \)
A common approach for such functional equations is to assume a polynomial form. Let's assume:
\[ f(x) = 1 - x^n \]
for some integer \( n \).

### Step 3: Verify the assumption
Now we need to check if this form satisfies the given condition. We calculate \( f\left(\frac{1}{x}\right) \):
\[ f\left(\frac{1}{x}\right) = 1 - \left(\frac{1}{x}\right)^n = 1 - \frac{1}{x^n} \]
Now, substituting \( f(x) \) and \( f\left(\frac{1}{x}\right) \) into the condition:
\[ (1 - x^n)\left(1 - \frac{1}{x^n}\right) = (1 - x^n) + \left(1 - \frac{1}{x^n}\right) \]
Expanding the left side:
\[ 1 - x^n - \frac{1}{x^n} + 1 = 2 - x^n - \frac{1}{x^n} \]
And the right side simplifies to:
\[ 2 - x^n - \frac{1}{x^n} \]
Both sides are equal, confirming our assumption is valid.

### Step 4: Use the condition \( f(3) = -26 \)
We know \( f(3) = -26 \):
\[ f(3) = 1 - 3^n = -26 \]
Solving for \( n \):
\[ 1 - 3^n = -26 \implies -3^n = -27 \implies 3^n = 27 \]
This gives:
\[ n = 3 \]

### Step 5: Find \( f(4) \)
Now we can find \( f(4) \):
\[ f(4) = 1 - 4^3 = 1 - 64 = -63 \]

### Conclusion
Thus, the value of \( f(4) \) is:
\[ \boxed{-63} \]