Question

If \(2x^3 + 3x^2 + 5x + 6 = 0\) has roots \(\alpha\), \(\beta\), \(\gamma\) then, find \(\alpha + \beta + \gamma\)

• A. 3/2
• B. -3/2
• C. 5/2
• D. -5/2

Hint:

Vieta's formula for sum of roots is universal:
\[ \text{Sum of roots} = -\frac{\text{Coefficient of second highest power}}{\text{Coefficient of highest power}} \]
Here, second highest power is \(x^2\) (coeff: 3) and highest power is \(x^3\) (coeff: 2).
Sum of roots = \(-\frac{3}{2}\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
For any polynomial equation, the relationships between the coefficients and the sums and products of its roots are defined by Vieta's formulas.
For a cubic equation of the form:
\[ ax^3 + bx^2 + cx + d = 0 \]
If the roots are \(\alpha\), \(\beta\), and \(\gamma\), the sum of the roots is given by:
\[ \alpha + \beta + \gamma = -\frac{b}{a} \]

Step 2: Detailed Explanation:
Let us identify the coefficients from the given cubic equation:
\[ 2x^3 + 3x^2 + 5x + 6 = 0 \]
Comparing this with the standard cubic form \(ax^3 + bx^2 + cx + d = 0\), we get:
- Coefficient of \(x^3\), \(a = 2\)
- Coefficient of \(x^2\), \(b = 3\)
- Coefficient of \(x\), \(c = 5\)
- Constant term, \(d = 6\)
Using Vieta's formula for the sum of the roots:
\[ \alpha + \beta + \gamma = -\frac{b}{a} \]
Substitute the values of \(a\) and \(b\):
\[ \alpha + \beta + \gamma = -\frac{3}{2} \]
Thus, the sum of the roots is \(-\frac{3}{2}\).
This matches Option (B).

Step 3: Final Answer:
The correct option is (B).