If `p + (1)/(p) = 112`, find `(p - 112)^(15) + (1)/(p^(15))`.

Options

A. 1
B. 15
C. 10
D. 0

Answer

To solve the equation \( p + \frac{1}{p} = 112 \) and find \( (p - 112)^{15} + \frac{1}{p^{15}} \), we can follow these steps:

### Step 1: Rearranging the Equation
We start with the equation:
\[ p + \frac{1}{p} = 112 \]
We can denote \( p - 112 \) as \( x \), so we have:
\[ p = x + 112 \]

### Step 2: Substitute \( p \) in the Original Equation
Substituting \( p \) in the original equation gives us:
\[ (x + 112) + \frac{1}{x + 112} = 112 \]
This simplifies to:
\[ x + 112 + \frac{1}{x + 112} = 112 \]
Subtracting 112 from both sides:
\[ x + \frac{1}{x + 112} = 0 \]

### Step 3: Isolate \( x \)
From the equation \( x + \frac{1}{x + 112} = 0 \), we can isolate \( x \):
\[ x = -\frac{1}{x + 112} \]
Multiplying both sides by \( x + 112 \) (assuming \( x + 112 \neq 0 \)):
\[ x(x + 112) = -1 \]
This expands to:
\[ x^2 + 112x + 1 = 0 \]

### Step 4: Solve the Quadratic Equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = 112, c = 1 \):
\[ x = \frac{-112 \pm \sqrt{112^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
Calculating the discriminant:
\[ 112^2 - 4 = 12544 - 4 = 12540 \]
So,
\[ x = \frac{-112 \pm \sqrt{12540}}{2} \]

### Step 5: Find \( p - 112 \)
Since we are interested in \( (p - 112)^{15} + \frac{1}{p^{15}} \), we note that:
\[ p - 112 = x \]
Thus, we need to find \( x^{15} + \frac{1}{p^{15}} \).

### Step 6: Calculate \( \frac{1}{p^{15}} \)
Using the identity \( p + \frac{1}{p} = 112 \), we can derive:
\[ p^{15} + \frac{1}{p^{15}} = (p + \frac{1}{p})^{15} - \text{(lower order terms)} \]
However, we can also use the earlier derived \( x + \frac{1}{x + 112} = 0 \) to see that:
\[ x^{15} + \frac{1}{p^{15}} = 0 \]

### Final Step: Conclusion
Thus, we conclude:
\[ (p - 112)^{15} + \frac{1}{p^{15}} = 0 \]