# Polynomials -finding sum of symmetric function of cubic polynomial

Problem :

If $\alpha , \beta,\gamma$ are the roots of $x^3+bx+c=0$ then $\alpha^2\beta +\alpha \beta^2+\beta^2\gamma +\beta \gamma^2+\gamma^2 \alpha+\gamma \alpha^2$ is equal to

Options are :

a) 3c

b) c

c) -c

d) -3c

Since : $\alpha + \beta +\gamma = \frac{-b}{a} = 0$ ; $\alpha \beta + \beta \gamma + \gamma \alpha = \frac{c}{a} =b$ ; $\alpha \beta \gamma = -c$

Please guide how to simplify the symmetric functions so we can use the above values : Thanks..

-

## 1 Answer

Hint: Try multiplying out $$(\alpha+\beta+\gamma)(\alpha\beta+\beta\gamma+\alpha\gamma)$$

-