If `x ^(2) -3x+2` is a factor of `x ^(4) -px ^(2) +q=0,` then `p+q=`

Answer

To solve the problem, we need to find the values of \( p \) and \( q \) given that \( x^2 - 3x + 2 \) is a factor of \( x^4 - px^2 + q = 0 \).

### Step-by-step Solution:

1. **Identify the Roots of the Factor**:
The quadratic \( x^2 - 3x + 2 \) can be factored as:
\[ (x - 1)(x - 2) = 0 \]
Thus, the roots are \( x = 1 \) and \( x = 2 \).

2. **Substitute the Roots into the Polynomial**:
Since \( x^2 - 3x + 2 \) is a factor of \( x^4 - px^2 + q \), both roots must satisfy the polynomial \( x^4 - px^2 + q = 0 \).

- For \( x = 1 \):
\[ 1^4 - p(1^2) + q = 0 \implies 1 - p + q = 0 \implies q = p - 1 \quad \text{(Equation 1)} \]

- For \( x = 2 \):
\[ 2^4 - p(2^2) + q = 0 \implies 16 - 4p + q = 0 \implies q = 4p - 16 \quad \text{(Equation 2)} \]

3. **Set the Equations Equal**:
Now we have two expressions for \( q \):
\[ p - 1 = 4p - 16 \]
Rearranging gives:
\[ -1 + 16 = 4p - p \implies 15 = 3p \implies p = 5 \]

4. **Find \( q \)**:
Substitute \( p = 5 \) back into Equation 1:
\[ q = 5 - 1 = 4 \]

5. **Calculate \( p + q \)**:
Now we can find \( p + q \):
\[ p + q = 5 + 4 = 9 \]

### Final Answer:
Thus, the value of \( p + q \) is \( \boxed{9} \).