If `a=x+1/x, " then "x^(3)+x^(-3)=`

Options

A. `a^(3)+3a`
B. `a^(3)-3a`
C. `a^(3)+3`
D. `a^(3)-3`

Answer

To solve the problem where \( a = x + \frac{1}{x} \) and we need to find \( x^3 + \frac{1}{x^3} \), we can follow these steps:

### Step 1: Identify the relationship
We know that:
\[ x^3 + \frac{1}{x^3} = \left( x + \frac{1}{x} \right) \left( x^2 - x \cdot \frac{1}{x} + \frac{1}{x^2} \right) \]
This can be simplified to:
\[ x^3 + \frac{1}{x^3} = \left( x + \frac{1}{x} \right) \left( x^2 - 1 + \frac{1}{x^2} \right) \]

### Step 2: Substitute \( a \)
Since \( a = x + \frac{1}{x} \), we can rewrite the expression as:
\[ x^3 + \frac{1}{x^3} = a \left( x^2 - 1 + \frac{1}{x^2} \right) \]

### Step 3: Find \( x^2 + \frac{1}{x^2} \)
To find \( x^2 + \frac{1}{x^2} \), we can use the identity:
\[ \left( x + \frac{1}{x} \right)^2 = x^2 + 2 + \frac{1}{x^2} \]
This implies:
\[ x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 = a^2 - 2 \]

### Step 4: Substitute back into the equation
Now we can substitute \( x^2 + \frac{1}{x^2} \) back into our equation:
\[ x^3 + \frac{1}{x^3} = a \left( (a^2 - 2) - 1 \right) = a (a^2 - 3) \]

### Step 5: Final expression
Thus, we have:
\[ x^3 + \frac{1}{x^3} = a^3 - 3a \]

### Conclusion
So, the final answer is:
\[ x^3 + \frac{1}{x^3} = a^3 - 3a \]