In the expression `(px ^(3) - 8x ^(2) - qx + 1),` then divisible by the expression `(3x ^(2) - 4x +1),` then what will be the vaue of p and q respectively ?

Options

A. `(21//4, 15, 8)`
B. `(6,1)`
C. `(33//4, 5//4)`
D. `(1,6)`

Answer

To solve the problem, we need to find the values of \( p \) and \( q \) such that the polynomial \( px^3 - 8x^2 - qx + 1 \) is divisible by \( 3x^2 - 4x + 1 \).

### Step-by-Step Solution:

1. **Identify the Roots of the Divisor**:
The expression \( 3x^2 - 4x + 1 \) can be factored or solved using the quadratic formula. The roots can be found using:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 3 \), \( b = -4 \), and \( c = 1 \).
\[ x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 3 \cdot 1}}{2 \cdot 3} = \frac{4 \pm \sqrt{16 - 12}}{6} = \frac{4 \pm 2}{6} \]
This gives us the roots:
\[ x = 1 \quad \text{and} \quad x = \frac{1}{3} \]

2. **Set Up Equations Using the Roots**:
Since \( px^3 - 8x^2 - qx + 1 \) must equal zero at these roots, we can substitute \( x = 1 \) and \( x = \frac{1}{3} \) into the polynomial.

- **For \( x = 1 \)**:
\[ p(1)^3 - 8(1)^2 - q(1) + 1 = 0 \]
Simplifying this gives:
\[ p - 8 - q + 1 = 0 \implies p - q - 7 = 0 \implies p - q = 7 \quad \text{(Equation 1)} \]

- **For \( x = \frac{1}{3} \)**:
\[ p\left(\frac{1}{3}\right)^3 - 8\left(\frac{1}{3}\right)^2 - q\left(\frac{1}{3}\right) + 1 = 0 \]
Simplifying this gives:
\[ \frac{p}{27} - \frac{8}{9} - \frac{q}{3} + 1 = 0 \]
Multiplying through by 27 to eliminate the fractions:
\[ p - 24 - 9q + 27 = 0 \implies p - 9q + 3 = 0 \implies p - 9q = -3 \quad \text{(Equation 2)} \]

3. **Solve the System of Equations**:
Now we have two equations:
- \( p - q = 7 \) (Equation 1)
- \( p - 9q = -3 \) (Equation 2)

We can solve these equations simultaneously. From Equation 1, we can express \( p \) in terms of \( q \):
\[ p = q + 7 \]
Substituting this into Equation 2:
\[ (q + 7) - 9q = -3 \]
Simplifying this:
\[ 7 - 8q = -3 \implies -8q = -3 - 7 \implies -8q = -10 \implies q = \frac{5}{4} \]

4. **Find \( p \)**:
Now substituting \( q \) back into Equation 1:
\[ p - \frac{5}{4} = 7 \implies p = 7 + \frac{5}{4} = \frac{28}{4} + \frac{5}{4} = \frac{33}{4} \]

### Final Values:
Thus, the values of \( p \) and \( q \) are:
\[ p = \frac{33}{4}, \quad q = \frac{5}{4} \]