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 two polynomials \( f(x) = x^3 - x^2 + px - 7 \) and \( g(x) = x^2 - 4x + q \) is \( (x - 1) \). This means that both polynomials must equal zero when \( x = 1 \).

### Step 1: Substitute \( x = 1 \) into \( f(x) \)

We start with the first polynomial:
\[ f(1) = 1^3 - 1^2 + p(1) - 7 \]
Calculating this gives:
\[ f(1) = 1 - 1 + p - 7 = p - 7 \]
Since \( (x - 1) \) is a divisor, we set \( f(1) = 0 \):
\[ p - 7 = 0 \implies p = 7 \]

### Step 2: Substitute \( x = 1 \) into \( g(x) \)

Next, we evaluate the second polynomial:
\[ g(1) = 1^2 - 4(1) + q \]
Calculating this gives:
\[ g(1) = 1 - 4 + q = q - 3 \]
Again, since \( (x - 1) \) is a divisor, we set \( g(1) = 0 \):
\[ q - 3 = 0 \implies q = 3 \]

### Step 3: Calculate \( p^2 + q^2 \)

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

### Final Answer

Thus, the value of \( p^2 + q^2 \) is:
\[ \boxed{58} \]