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Math Help - Symmetric properties of the roots of polynomial equations

  1. #1
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    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:

    Ax + Bx + 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 : x - (sum)x + (pairs)x - (product) = 0


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

    Thanks =)
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  2. #2
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    Quote Originally Posted by djmccabie View Post
    Symmetric properties of the roots of polynomial equations

    Cubics

    Original Equation:

    Ax + Bx + 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 : x - (sum)x + (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.
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