Solution to general cubic equation

Solution to general cubic equation

This is in continuance to my earlier post : Deriving expression for general cubic equation solution. Also, the links are repeated for the first four pages of the book by Dickson, titled Introduction to Theory of Algebraic Equations are given as : page 1,,page 2,, page 3, ,page 4 On page 4, the second question asks to show : The cubic 2, i.e. the reduced cubic equation: y3pyq0, with p c_2-frac13c_12,,,,,,,,,,, q -c_3frac13c_1c_2 - frac227c_13 R frac14q2 frac127p3frac14-c_3frac13c_1c_2 - frac227c_132 frac127c_2-frac13c_123 frac14c_32 frac19c_12c_22 frac4272c_16 - frac481c_14c_2-frac23c_1c_2c_3frac427c_13c_3frac127c_23frac127c_16-c_22c_12frac13c_2c_14 frac14c_32 frac136c_12c_22 frac1272c_16-frac181c_14c_2 -frac16c_1c_2c_3 frac127c_13c_3frac127c_23frac1272c_16-frac127c_22c_12 frac181c_2c_14 frac14c_32frac136c_12c_22frac2272c_16-frac16c_1c_2c_3frac127c_13c_3frac127c_23-frac127c_22c_13 has :i one real root and two imaginary roots if Rgt 0, i three real root, two of which are equal R 0, i three real distinct roots irreducible case if Rlt 0

This is generally not true for p,q in mathbbC. For example, consider the polynomial y3i0. Then p0, qi, and Rfrac14q2frac127p3-frac14 0 but it has no real root. However, this is true for p,q in mathbbR. If p,q in mathbbR, we have at least one real root and any complex roots comes in complex conjugate pair. beginaligny_1-y_22y_2-y_32y_3-y_12-27q2-4p3-27q24p3-427left frac14q2frac127p3right-427R endalign If R0, then y_1-y_22y_2-y_32y_3-y_120, hence we must have an imaginary root. Hence we have one real root and two imaginary roots. If R0, then we must have at least a pair of repeating root as y_1-y_22y_2-y_32y_3-y_120 If R0, then y_1-y_22y_2-y_32y_3-y_120. Lets rule out the possibility that we have any complex root. Suppose y_2, y_3 are complex conjugate. Let y_1 be the real root, let y_2 alpha beta i and y_3 alpha - beta i where alpha, beta in mathbbR, beta ne 0. Then y_1-y_22y_2-y_32y_3-y_12y_1-alpha-beta i22beta i2alpha-beta i -y_120 which is a contradicition. Hence if R0, every root is distinct.