# Symmetric properties of the roots of polynomial equations

• Sep 30th 2008, 10:53 AM
djmccabie
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 =)
• Sep 30th 2008, 11:14 AM
Opalg
Quote:

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.