If `x+(1)/(x)=17`, what is the value of `(x^(4)+(1)/(x^(2)))/(x^(2)-3x+1)`?

Options

A. `(2431)/(7)`
B. `(3375)/(7)`
C. `(3375)/(14)`
D. `(3985)/(9)`

Answer

To solve the problem step by step, we start with the equation given:

1. **Given Equation**:
\[ x + \frac{1}{x} = 17 \]

2. **Square Both Sides**:
We can square both sides of the equation to find \(x^2 + \frac{1}{x^2}\):
\[ \left(x + \frac{1}{x}\right)^2 = 17^2 \]
Expanding the left side:
\[ x^2 + 2 + \frac{1}{x^2} = 289 \]
Therefore, we have:
\[ x^2 + \frac{1}{x^2} = 289 - 2 = 287 \]

3. **Find \(x^4 + \frac{1}{x^4}\)**:
Now we can find \(x^4 + \frac{1}{x^4}\) using the identity:
\[ x^4 + \frac{1}{x^4} = \left(x^2 + \frac{1}{x^2}\right)^2 - 2 \]
Substituting the value we found:
\[ x^4 + \frac{1}{x^4} = 287^2 - 2 \]
First, calculate \(287^2\):
\[ 287^2 = 82369 \]
Thus:
\[ x^4 + \frac{1}{x^4} = 82369 - 2 = 82367 \]

4. **Find \(x^2 - 3x + 1\)**:
We need to evaluate \(x^2 - 3x + 1\). We already have \(x + \frac{1}{x} = 17\), which gives us \(x = 17 - \frac{1}{x}\). To find \(x^2\), we can use the equation:
\[ x^2 = (x + \frac{1}{x})^2 - 2 = 17^2 - 2 = 289 - 2 = 287 \]
Now substituting \(x\) into \(x^2 - 3x + 1\):
\[ x^2 - 3x + 1 = 287 - 3(17) + 1 \]
Calculate \(3(17)\):
\[ 3(17) = 51 \]
Thus:
\[ x^2 - 3x + 1 = 287 - 51 + 1 = 237 \]

5. **Final Calculation**:
Now we can substitute back into our original expression:
\[ \frac{x^4 + \frac{1}{x^4}}{x^2 - 3x + 1} = \frac{82367}{237} \]

6. **Perform the Division**:
Performing the division gives:
\[ \frac{82367}{237} = 347 \]

Thus, the final answer is:
\[ \boxed{347} \]