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 1: Find the roots of the factor
The first step is to find the roots of the quadratic equation \( x^2 - 3x + 2 = 0 \).

We can factor this quadratic:
\[ x^2 - 3x + 2 = (x - 2)(x - 1) = 0 \]
Thus, the roots are \( x = 1 \) and \( x = 2 \).

### Step 2: Substitute the roots into the quartic equation
Since \( x^2 - 3x + 2 \) is a factor of \( x^4 - px^2 + q \), both roots must satisfy the quartic equation \( x^4 - px^2 + q = 0 \).

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

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

### Step 3: Solve the system of equations
Now we have a system of two equations:
1. \( p - q = 1 \) (Equation 1)
2. \( 4p - q = 16 \) (Equation 2)

We can solve these equations simultaneously. From Equation 1, we can express \( q \) in terms of \( p \):
\[ q = p - 1 \]

Now substitute \( q \) in Equation 2:
\[ 4p - (p - 1) = 16 \]
Simplifying this gives:
\[ 4p - p + 1 = 16 \implies 3p + 1 = 16 \implies 3p = 15 \implies p = 5 \]

### Step 4: Find \( q \)
Now that we have \( p \), we can find \( q \):
\[ q = p - 1 = 5 - 1 = 4 \]

### Step 5: Calculate \( p + q \)
Finally, we can find \( p + q \):
\[ p + q = 5 + 4 = 9 \]

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