$ ext{If } \alpha, \beta, \gamma ext{ are the zeros of the cubic polynomial } ax^3 + bx^2 + cx + d, ext{ then } \alpha\beta + \beta\gamma + \gamma\alpha ext{ is equal to : }$

Q: $\text{If } \alpha, \beta, \gamma \text{ are the zeros of the cubic polynomial } ax^3 + bx^2 + cx + d, \text{ then } \alpha\beta + \beta\gamma + \gamma\alpha \text{ is equal to : }$

To determine the value of $\alpha\beta + \beta\gamma + \gamma\alpha$ for the cubic polynomial $ax^3 + bx^2 + cx + d$ with roots $\alpha, \beta, \gamma$, we can use Vieta's formulas. Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. For a cubic polynomial $ax^3 + bx^2 + cx + d$, Vieta's formulas give us the following relationships: 1. $\alpha + \beta + \gamma = -\frac{b}{a}$ 2. $\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}$ 3. $\alpha\beta\gamma = -\frac{d}{a}$ According to Vieta's formulas, the expression $\alpha\beta + \beta\gamma + \gamma\alpha$ is equal to $\frac{c}{a}$. Therefore, the correct answer is Option 3. Now, let's explain why each of the other options is incorrect: 1. **Option 1: $-\frac{b}{a}$** - This option represents the sum of the roots, $\alpha + \beta + \gamma$, not the sum of the products of the roots taken two at a time. Therefore, it is incorrect for $\alpha\beta + \beta\gamma + \gamma\alpha$. 2. **Option 2: $\frac{b}{a}$** - This option does not correspond to any of the standard expressions derived from Vieta's formulas for a cubic polynomial. It is neither the sum of the roots nor the sum of the products of the roots taken two at a time. Thus, it is incorrect. 3. **Option 4: $\frac{d}{a}$** - This option does not relate to any of the expressions for the roots of the polynomial according to Vieta's formulas. Specifically, it does not represent the sum of the products of the roots taken two at a time. Therefore, it is incorrect. In conclusion, based on Vieta's formulas, the correct expression for $\alpha\beta + \beta\gamma + \gamma\alpha$ is $\frac{c}{a}$, which is why Option 3 is the correct answer.

Question

$	ext{If } \alpha, \beta, \gamma 	ext{ are the zeros of the cubic polynomial } ax^3 + bx^2 + cx + d, 	ext{ then } \alpha\beta + \beta\gamma + \gamma\alpha 	ext{ is equal to : }$

Accepted Answer

To determine the value of $\alpha\beta + \beta\gamma + \gamma\alpha$ for the cubic polynomial $ax^3 + bx^2 + cx + d$ with roots $\alpha, \beta, \gamma$, we can use Vieta's formulas. Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.

For a cubic polynomial $ax^3 + bx^2 + cx + d$, Vieta's formulas give us the following relationships:

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

According to Vieta's formulas, the expression $\alpha\beta + \beta\gamma + \gamma\alpha$ is equal to $\frac{c}{a}$. Therefore, the correct answer is Option 3.

Now, let's explain why each of the other options is incorrect:

1. **Option 1: $-\frac{b}{a}$** - This option represents the sum of the roots, $\alpha + \beta + \gamma$, not the sum of the products of the roots taken two at a time. Therefore, it is incorrect for $\alpha\beta + \beta\gamma + \gamma\alpha$.

2. **Option 2: $\frac{b}{a}$** - This option does not correspond to any of the standard expressions derived from Vieta's formulas for a cubic polynomial. It is neither the sum of the roots nor the sum of the products of the roots taken two at a time. Thus, it is incorrect.

3. **Option 4: $\frac{d}{a}$** - This option does not relate to any of the expressions for the roots of the polynomial according to Vieta's formulas. Specifically, it does not represent the sum of the products of the roots taken two at a time. Therefore, it is incorrect.

In conclusion, based on Vieta's formulas, the correct expression for $\alpha\beta + \beta\gamma + \gamma\alpha$ is $\frac{c}{a}$, which is why Option 3 is the correct answer.