Mathematics - Quadratic Equation Question with Solution | TestHub

Since $\alpha$, $\beta$, and $\gamma$ are the roots of $x^3+px+q=0$, we know that the sum of the roots is zero:

$$\alpha +\beta +\gamma =0$$

Now, consider the given determinant:

$$\left|\begin{array}{ccc}\alpha +\beta +\gamma & \beta & \gamma \\ \alpha +\beta +\gamma & \gamma & \alpha \\ \alpha +\beta +\gamma & \alpha & \beta \end{array}\right|$$

Applying the column operation $C_1\to C_1+C_2+C_3$, the determinant becomes:

$$\left|\begin{array}{ccc}0& \beta & \gamma \\ 0& \gamma & \alpha \\ 0& \alpha & \beta \end{array}\right|=0$$

Since the first column contains all zeros, the value of the determinant is 0.