[Math] If $\alpha, \beta, \gamma$ are the roots of the cubic $\ ax^3+bx^2+cx+d$, show that $\alpha\beta+\alpha\gamma+\beta\gamma=\frac{c}{a}$.

Question

I know that $\alpha + \beta + \gamma = -\frac{b}{a}%$ and that $\alpha\beta\gamma=-\frac{d}{a}$, but I don't know how to go from there to $\alpha\beta+\alpha\gamma+\beta\gamma=\frac{c}{a}$.

Answer 1

Hint:

Notice that we have $$ax^3+bx^2+cx+d=a(x-\alpha)(x-\beta)(x-\gamma)$$