If `p=x+1/x` then the value of `p-1/p` will be

Options

A. 3x
B. `3/x`
C. `(x^4+x^2 +1)/(x^3+x)`
D. `(x^4+3x^2 + 1)/(x^3+x)`

Answer

To solve the problem, we need to find the value of \( p - \frac{1}{p} \) given that \( p = x + \frac{1}{x} \).

### Step-by-step Solution:

1. **Substituting the value of \( p \)**:
\[ p = x + \frac{1}{x} \]

2. **Finding \( \frac{1}{p} \)**:
To find \( \frac{1}{p} \), we take the reciprocal of \( p \):
\[ \frac{1}{p} = \frac{1}{x + \frac{1}{x}} = \frac{x}{x^2 + 1} \]
(This is derived by multiplying numerator and denominator by \( x \) to eliminate the fraction in the denominator.)

3. **Calculating \( p - \frac{1}{p} \)**:
Now we need to compute \( p - \frac{1}{p} \):
\[ p - \frac{1}{p} = \left( x + \frac{1}{x} \right) - \frac{x}{x^2 + 1} \]

4. **Finding a common denominator**:
The common denominator for the terms is \( x(x^2 + 1) \):
\[ p - \frac{1}{p} = \frac{(x + \frac{1}{x})(x^2 + 1) - x}{x(x^2 + 1)} \]

5. **Expanding the numerator**:
\[ = \frac{(x^3 + x) + (1) - x}{x(x^2 + 1)} \]
Simplifying the numerator:
\[ = \frac{x^3 + 1}{x(x^2 + 1)} \]

6. **Final expression**:
Thus, we have:
\[ p - \frac{1}{p} = \frac{x^3 + 1}{x(x^2 + 1)} \]