Symmetric properties of the roots of polynomial equations

Hi just had this plonked on me and feel it would make better sense if someone could explain it to me :)

__Symmetric properties of the roots of polynomial equations__

Cubics

Original Equation:

Ax³ + Bx² + Cx + D = 0

Sum of roots : α + β + γ = -B/A

'Roots in pairs' : βγ + γα + αβ = C/A

Product of the roots : αβγ = -D/A

New equation : x³ - (sum)x² + (pairs)x - (product) = 0

I could memorise all this but would rather know what it means and how it works.

Thanks =)