Question

If $\alpha, \beta, \gamma$ are the zeroes of $4x^3 + 8x^2 - 6x - 2$, then the value of $\alpha\beta + \beta\gamma + \gamma\alpha$ is _____.

• A. $ \frac{2}{3} $
• B. $ -\frac{2}{3} $
• C. $ \frac{3}{2} $
• D. $ -\frac{3}{2} $

Hint:

For any cubic polynomial: \[ ax^3+bx^2+cx+d \] always remember: \[ \alpha+\beta+\gamma=-\frac{b}{a} \] \[ \alpha\beta+\beta\gamma+\gamma\alpha=\frac{c}{a} \] \[ \alpha\beta\gamma=-\frac{d}{a} \] The sign pattern is very important in exams, so memorize it carefully.

Answer 1

The Correct Option is D

Concept: For a cubic polynomial of the form \[ ax^3 + bx^2 + cx + d \] having zeroes $\alpha$, $\beta$, and $\gamma$, the relationships between coefficients and zeroes are: \[ \alpha + \beta + \gamma = -\frac{b}{a} \] \[ \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} \] \[ \alpha\beta\gamma = -\frac{d}{a} \] In this problem, we only need the second relation.

Step 1: Compare the given polynomial with the standard form. The given polynomial is \[ 4x^3 + 8x^2 - 6x - 2 \] Comparing with \[ ax^3 + bx^2 + cx + d \] we get: \[ a = 4 \] \[ b = 8 \] \[ c = -6 \] \[ d = -2 \]

Step 2: Use the formula for the sum of products of zeroes taken two at a time. We know that \[ \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} \] Substituting the values: \[ \alpha\beta + \beta\gamma + \gamma\alpha = \frac{-6}{4} \]

Step 3: Simplify the fraction carefully. Both numerator and denominator are divisible by 2: \[ \frac{-6}{4} = \frac{-3}{2} \] Therefore, \[ \boxed{\alpha\beta + \beta\gamma + \gamma\alpha = -\frac{3}{2}} \] Hence, the correct option is \[ \boxed{(4)\ -\frac{3}{2}} \]