# Thread: Symmetric properties of the roots of polynomial equations

1. ## 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 =)

2. Originally Posted by djmccabie
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.
If the roots are α, β and γ then it follows from the factor theorem that the equation must be A(x-α)(x-β)(x-γ) = 0. If you multiply this out and compare it with the original equation then you'll see why those formulas hold.