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

Options

A. a) (-4, -5)
B. b) (4, 5)
C. c) (-5, 4)
D. d) (5, 4)

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: Factor the quadratic expression
First, we factor the quadratic expression \( x^2 - 3x + 2 \).

The factors of the quadratic can be found using the middle-term splitting method or by factoring directly:
\[ x^2 - 3x + 2 = (x - 1)(x - 2) \]

### Step 2: Identify the roots
The roots of the equation \( x^2 - 3x + 2 = 0 \) are:
\[ x = 1 \quad \text{and} \quad x = 2 \]

### Step 3: 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 \( x^4 - px^2 + q = 0 \).

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

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

### Step 4: Set the equations equal to each other
Now we have two equations for \( q \):
1. \( q = p - 1 \)
2. \( q = 4p - 16 \)

Setting them equal to each other:
\[ p - 1 = 4p - 16 \]

### Step 5: Solve for \( p \)
Rearranging the equation:
\[ -1 + 16 = 4p - p \implies 15 = 3p \implies p = 5 \]

### Step 6: Substitute \( p \) back to find \( q \)
Now substitute \( p = 5 \) back into Equation 1:
\[ q = 5 - 1 = 4 \]

### Final Answer
Thus, the values of \( p \) and \( q \) are:
\[ (p, q) = (5, 4) \]