Question

If α , β and γ are the zeroes of the cubic polynomial \(2x^3+x^2-13x+6\) , then the value of \(αβγ\) is

• A. 3
• B. -3
• C. \(\frac{-1}{2}\)
• D. \(\frac{-13}{2}\)

Answer 1

The Correct Option is B

The given cubic polynomial is \(2x^3 + x^2 - 13x + 6\). 
The roots (zeroes) of this polynomial are denoted by \(\alpha, \beta, \gamma\). 
According to Viète's formulas, for a cubic polynomial \(ax^3 + bx^2 + cx + d = 0\), the relations between the coefficients and the roots are: 

1. \(\alpha + \beta + \gamma = -\frac{b}{a}\)
2. \(\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}\)
3. \(\alpha\beta\gamma = -\frac{d}{a}\)

Here, \(a = 2\), \(b = 1\), \(c = -13\), and \(d = 6\). 
We need to find the value of \(\alpha\beta\gamma\): 
\[ \alpha\beta\gamma = -\frac{d}{a} = -\frac{6}{2} = -3 \]
Hence, the value of \(\alpha\beta\gamma\) is \(-3\).

Answer 2

We are given the cubic polynomial:

\[ P(x) = 2x^3 + x^2 - 13x + 6 \]

Step 1: Identify the product of the roots 

For a cubic polynomial of the form:

\[ ax^3 + bx^2 + cx + d \]

The product of the roots (\( \alpha \beta \gamma \)) is given by:

\[ \alpha \beta \gamma = \frac{-d}{a} \]

Here, \( a = 2 \) and \( d = 6 \), so:

\[ \alpha \beta \gamma = \frac{-6}{2} = -3 \]

Final Answer: \( -3 \).