If greatest common divisor of `x^(3)-x^(2)+px - 7` and `x^(2)-4x +q` is `(x-1)`, then the value of `p^(2)+q^(2)` is

Options

A. 58
B. 85
C. 75
D. 57

Answer

To solve the problem, we need to find the values of \( p \) and \( q \) such that the greatest common divisor (GCD) of the polynomials \( x^3 - x^2 + px - 7 \) and \( x^2 - 4x + q \) is \( x - 1 \).

### Step-by-Step Solution:

1. **Understanding the GCD Condition**:
Since \( x - 1 \) is the GCD of the two polynomials, both polynomials must equal zero when \( x = 1 \). This means that substituting \( x = 1 \) into both polynomials should yield zero.

2. **Substituting in the First Polynomial**:
The first polynomial is \( x^3 - x^2 + px - 7 \).
- Substitute \( x = 1 \):
\[ 1^3 - 1^2 + p(1) - 7 = 0 \]
Simplifying this gives:
\[ 1 - 1 + p - 7 = 0 \implies p - 7 = 0 \implies p = 7 \]

3. **Substituting in the Second Polynomial**:
The second polynomial is \( x^2 - 4x + q \).
- Substitute \( x = 1 \):
\[ 1^2 - 4(1) + q = 0 \]
Simplifying this gives:
\[ 1 - 4 + q = 0 \implies q - 3 = 0 \implies q = 3 \]

4. **Calculating \( p^2 + q^2 \)**:
Now that we have \( p = 7 \) and \( q = 3 \), we can calculate \( p^2 + q^2 \):
\[ p^2 + q^2 = 7^2 + 3^2 = 49 + 9 = 58 \]

5. **Final Answer**:
Therefore, the value of \( p^2 + q^2 \) is \( \boxed{58} \).