(11C) CUBIC EQUATION

07202021, 10:12 PM
(This post was last modified: 09142021 05:36 PM by Albert Chan.)
Post: #21




RE: (11C) CUBIC EQUATION
(06102018 07:15 AM)Dieter Wrote: So it can't be the discriminant of the original cubic equation (here D<0 means three real roots). It is interesting cubic discriminant sign test is opposite of quadratic (cubic Δ < 0 → 3 real and unequal roots) We can show the reason with identity: x³ + y³ + z³ − 3xyz = (x+y+z) * (x+yω+z/ω) * (x+y/ω+zω), where ω = e^(i*2*pi/3) LHS is a depressed cubic: x³ + p*x + q , where p = 3yz, q = y³ + z³ We then setup a quadratic of t, with roots (y³, z³) (t  y³)*(t  z³) = t²  q*t  (p/3)³ → t = q/2 ± √Δ, where Δ = (q/2)² + (p/3)³ Defined this way, cubic discriminant is really quadratic discriminant (*) If Δ < 0, t is complex, (y,z) is conjugate of each other, y + z = y + conj(y) = 2*Re(y) All 3 roots are thus real and unequal: (y = ³√t , z = (p/3)/y) Δ < 0: x³ + p*x + q = (x + 2*Re(y)) * (x + 2*Re(yω)) * (x + 2*Re(y/ω)) (*) CRC Handbook of Mathematical Science 6th ed. (Beyer) also define cubic Δ this way. But, Δ is likely defined scaled by 108, to (4p³+27q²), matching General formula of discriminant 

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