Galois Theory: Finding the discriminant of a polynomial

Question

I am very confused on how to determine the coefficient of: $f(x)=x^3-s_1x^2+s_2x-s_3$. Artin is really proving me with too much abstraction, is there an easier way to be thinking about how to do this. I need to somehow get a relation between this polynomial and it expressed with its actual roots.

Answer 1

Assume the roots $\alpha_1,\alpha_2, \alpha_3$, then you will have the equality

$$ (x-\alpha_1)(x-\alpha_2)(x-\alpha_3)=x^3-s_1x^2+s_2x-s_3.$$

Now, work out the left hand side of the above equation and try to finish the problem.