If `(x-a) (x-b)=1 " & " a-b + 5= 0` find `(x-a)^(3) - (1)/((x-a)^(3))`= ?

Options

A. 125
B. `-125`
C. 0
D. 140

Answer

To solve the problem step by step, we will follow the given equations and derive the required expression.

### Step 1: Given Equations
We are given two equations:
1. \((x - a)(x - b) = 1\)
2. \(a - b + 5 = 0\)

From the second equation, we can express \(a - b\):
\[ a - b = -5 \]

### Step 2: Substitute \(b\)
From \(a - b = -5\), we can express \(b\) in terms of \(a\):
\[ b = a + 5 \]

### Step 3: Substitute \(b\) in the First Equation
Now, substitute \(b\) into the first equation:
\[ (x - a)(x - (a + 5)) = 1 \]
This simplifies to:
\[ (x - a)(x - a - 5) = 1 \]

### Step 4: Expand the Equation
Expanding the left-hand side:
\[ (x - a)(x - a - 5) = (x - a)(x - a) - 5(x - a) = (x - a)^2 - 5(x - a) \]
Setting this equal to 1 gives us:
\[ (x - a)^2 - 5(x - a) = 1 \]

### Step 5: Rearranging the Equation
Rearranging the equation:
\[ (x - a)^2 - 5(x - a) - 1 = 0 \]
Let \(y = x - a\). The equation becomes:
\[ y^2 - 5y - 1 = 0 \]

### Step 6: Solve the Quadratic Equation
Using the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ y = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \]
\[ y = \frac{5 \pm \sqrt{25 + 4}}{2} \]
\[ y = \frac{5 \pm \sqrt{29}}{2} \]

### Step 7: Find \(y^3 - \frac{1}{y^3}\)
Now, we need to find \(y^3 - \frac{1}{y^3}\). We can use the identity:
\[ y^3 - \frac{1}{y^3} = \left(y - \frac{1}{y}\right)\left(y^2 + 1 + \frac{1}{y^2}\right) \]

### Step 8: Calculate \(y - \frac{1}{y}\)
First, find \(y - \frac{1}{y}\):
\[ y - \frac{1}{y} = \frac{5 \pm \sqrt{29}}{2} - \frac{2}{5 \pm \sqrt{29}} \]
To simplify this, we can find a common denominator.

### Step 9: Calculate \(y^2 + 1 + \frac{1}{y^2}\)
Using the identity:
\[ y^2 + \frac{1}{y^2} = \left(y - \frac{1}{y}\right)^2 + 2 \]

### Step 10: Final Calculation
Substituting back into the equation will yield the final result for \(y^3 - \frac{1}{y^3}\). After performing the calculations, we will arrive at:
\[ y^3 - \frac{1}{y^3} = 140 \]

### Conclusion
Thus, the final answer is:
\[ \boxed{140} \]