If the roots of `x^(3) - 42x^(2) + 336x - 512 = 0 `, are in increasing geometric progression, its common ratio is

Options

A. a) 2
B. b) 3
C. c) 4
D. d) 6

Answer

To solve the problem, we need to find the common ratio of the roots of the polynomial equation \(x^3 - 42x^2 + 336x - 512 = 0\) given that the roots are in increasing geometric progression (GP).

### Step-by-Step Solution:

1. **Identify the Roots in GP**:
Let the roots be \(A\), \(Ar\), and \(Ar^2\), where \(A\) is the first term and \(r\) is the common ratio.

2. **Sum of the Roots**:
According to Vieta's formulas, the sum of the roots is equal to the coefficient of \(x^2\) with the opposite sign. Therefore:
\[ A + Ar + Ar^2 = 42 \]
This can be factored as:
\[ A(1 + r + r^2) = 42 \quad \text{(Equation 1)} \]

3. **Product of the Roots**:
The product of the roots is equal to the constant term (with the opposite sign) divided by the leading coefficient. Thus:
\[ A \cdot Ar \cdot Ar^2 = 512 \]
This simplifies to:
\[ A^3 r^3 = 512 \]
Taking the cube root gives:
\[ Ar = 8 \quad \text{(Equation 2)} \]

4. **Substituting \(A\) in Equation 1**:
From Equation 2, we can express \(A\) in terms of \(r\):
\[ A = \frac{8}{r} \]
Substituting this into Equation 1:
\[ \frac{8}{r}(1 + r + r^2) = 42 \]
Multiplying through by \(r\):
\[ 8(1 + r + r^2) = 42r \]
Expanding gives:
\[ 8 + 8r + 8r^2 = 42r \]
Rearranging leads to:
\[ 8r^2 - 34r + 8 = 0 \quad \text{(Equation 3)} \]

5. **Solving the Quadratic Equation**:
We can simplify Equation 3 by dividing through by 2:
\[ 4r^2 - 17r + 4 = 0 \]
Using the quadratic formula \(r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ r = \frac{17 \pm \sqrt{(-17)^2 - 4 \cdot 4 \cdot 4}}{2 \cdot 4} \]
\[ r = \frac{17 \pm \sqrt{289 - 64}}{8} \]
\[ r = \frac{17 \pm \sqrt{225}}{8} \]
\[ r = \frac{17 \pm 15}{8} \]
This gives us two possible values for \(r\):
\[ r = \frac{32}{8} = 4 \quad \text{and} \quad r = \frac{2}{8} = \frac{1}{4} \]

6. **Choosing the Correct Common Ratio**:
Since the roots are in increasing geometric progression, we need \(r\) to be greater than 1. Thus, we select:
\[ r = 4 \]

### Final Answer:
The common ratio \(r\) is \(4\).