If  α, β  and  γ  are the zeros of the polynomial  2x3 − 6x2 − 4x + 30 . then the value of  (αβ + βγ + γα)  isSelect an answerA–2B2C5D–30

1. Break Down the Problem

We need to find the sum of the products of the roots taken two at a time, which is given by the expression $\alpha\beta + \beta\gamma + \gamma\alpha$ for the polynomial $2x^3 - 6x^2 - 4x + 30$.

2. Relevant Concepts

For a cubic polynomial in the form: $ax^3 + bx^2 + cx + d = 0$ the relationships among the roots (α, β, γ) can be determined using Vieta's formulas:

• Sum of the roots: $\alpha + \beta + \gamma = -\frac{b}{a}$
• Sum of the product of the roots taken two at a time: $\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}$
• Product of the roots: $\alpha\beta\gamma = -\frac{d}{a}$

For our polynomial:

• $a = 2$
• $b = -6$
• $c = -4$
• $d = 30$

3. Analysis and Detail

Using Vieta's formulas:

1. The sum of the roots is: $\alpha + \beta + \gamma = -\frac{-6}{2} = \frac{6}{2} = 3$
2. The sum of the products of the roots taken two at a time is: $\alpha\beta + \beta\gamma + \gamma\alpha = \frac{-4}{2} = -2$
3. The product of the roots is: $\alpha\beta\gamma = -\frac{30}{2} = -15$

4. Verify and Summarize

We found the required value: $\alpha\beta + \beta\gamma + \gamma\alpha = -2$ which corresponds to option A.

Final Answer

The value of $\alpha\beta + \beta\gamma + \gamma\alpha$ is $\boxed{-2}$.