Theory of Equation

Ivanhy92

Let a/k, a, and ak be the roots of the equation x^3+px^2+qx+r=0, where k≠0
From the relations of roots and coefficients,
a/k + a + ak = -p                => a(1/k + 1 + k) = -p --------(1)
a^2/k + a^2 + (a^2)k = q  => (a^2)(1/k + 1 + k) = q -----(2)
a^3 = -r                            =>  a = (-r)^(1/3) -------------(3)
(2)/(1) : a = q/(-p)
Put (3) into above,
(-r)^(1/3) = q/(-p)
-r = (q^3)/(-p)^3
∴(p^3)r = (q^3)

For 3x^3 - 26x^2 + 52x - 24=0,
x^3 - (26/3)x^2 + (52/3)x - 8 = 0
∵[-(26/3)]^3 x 8 = -140608/27 = (-52/3)^3
∴ By (3), a = 8^(1/3) = 2
Put a=2 into (1),
2(1/k + 1 + k) = 26/3
k = ……