If p, q are the zeroes of the polynomial f(x) = `x^(2)` + k(x - 1) - c, then (p - 1)(q- 1) is equal to ______.

Options

A. c-1
B. 1 -c
C. c
D. 1 +c

Answer

To solve the problem, we need to find the value of \((p - 1)(q - 1)\) where \(p\) and \(q\) are the zeroes of the polynomial \(f(x) = x^2 + k(x - 1) - c\).

### Step-by-Step Solution:

1. **Rewrite the Polynomial**:
The given polynomial is:
\[ f(x) = x^2 + k(x - 1) - c \]
We can expand this:
\[ f(x) = x^2 + kx - k - c \]
So, the polynomial can be rewritten as:
\[ f(x) = x^2 + kx - (k + c) \]

2. **Identify Coefficients**:
From the polynomial \(f(x) = x^2 + kx - (k + c)\), we can identify:
- Coefficient of \(x^2\) (A) = 1
- Coefficient of \(x\) (B) = k
- Constant term (C) = \(-(k + c)\)

3. **Sum and Product of Zeroes**:
For a quadratic polynomial \(ax^2 + bx + c\):
- The sum of the zeroes \(p + q\) is given by:
\[ p + q = -\frac{B}{A} = -\frac{k}{1} = -k \]
- The product of the zeroes \(pq\) is given by:
\[ pq = \frac{C}{A} = \frac{-(k + c)}{1} = -(k + c) \]

4. **Finding \((p - 1)(q - 1)\)**:
We can express \((p - 1)(q - 1)\) using the identity:
\[ (p - 1)(q - 1) = pq - (p + q) + 1 \]
Substituting the values we found:
\[ (p - 1)(q - 1) = pq - (p + q) + 1 \]
\[ = [-(k + c)] - (-k) + 1 \]
\[ = -(k + c) + k + 1 \]
\[ = -c + 1 \]

5. **Final Result**:
Therefore, the value of \((p - 1)(q - 1)\) is:
\[ = 1 - c \]

### Conclusion:
The answer is:
\[ \boxed{1 - c} \]