[CAES] (x^4-4*x^3+2*x^2+4*x+4) Exact factorization of Quartic Polynomial

[Tim Wessman]

(04-30-2016 08:40 PM)compsystems Wrote:  symbolic solution


https://www.wolframalpha.com/input/?i=so...2B+e+%3D+0

I think you have just conclusively demonstrated exactly why this is not implemented, and validated what everyone else in this thread is saying.

Here is *one* of your roots. There are four more.

Quote:(x == -b/(4 a) - Sqrt[b^2/(4 a^2) - (2 c)/(3 a) + (2^(1/3) (c^2 - 3 b d + 12 a e))/(3 a (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)) + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)/(3 2^(1/3) a)]/2 - Sqrt[b^2/(2 a^2) - (4 c)/(3 a) - (2^(1/3) (c^2 - 3 b d + 12 a e))/(3 a (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)) - (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)/(3 2^(1/3) a) - (-(b^3/a^3) + (4 b c)/a^2 - (8 d)/a)/(4 Sqrt[b^2/(4 a^2) - (2 c)/(3 a) + (2^(1/3) (c^2 - 3 b d + 12 a e))/(3 a (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)) + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)/(3 2^(1/3) a)])]/2 && a != 0) || (x == -b/(4 a) - Sqrt[b^2/(4 a^2) - (2 c)/(3 a) + (2^(1/3) (c^2 - 3 b d + 12 a e))/(3 a (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)) + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)/(3 2^(1/3) a)]/2 + Sqrt[b^2/(2 a^2) - (4 c)/(3 a) - (2^(1/3) (c^2 - 3 b d + 12 a e))/(3 a (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)) - (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)/(3 2^(1/3) a) - (-(b^3/a^3) + (4 b c)/a^2 - (8 d)/a)/(4 Sqrt[b^2/(4 a^2) - (2 c)/(3 a) + (2^(1/3) (c^2 - 3 b d + 12 a e))/(3 a (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)) + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)/(3 2^(1/3) a)])]/2 && a != 0) || (x == -b/(4 a) + Sqrt[b^2/(4 a^2) - (2 c)/(3 a) + (2^(1/3) (c^2 - 3 b d + 12 a e))/(3 a (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)) + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)/(3 2^(1/3) a)]/2 - Sqrt[b^2/(2 a^2) - (4 c)/(3 a) - (2^(1/3) (c^2 - 3 b d + 12 a e))/(3 a (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)) - (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)/(3 2^(1/3) a) + (-(b^3/a^3) + (4 b c)/a^2 - (8 d)/a)/(4 Sqrt[b^2/(4 a^2) - (2 c)/(3 a) + (2^(1/3) (c^2 - 3 b d + 12 a e))/(3 a (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)) + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)/(3 2^(1/3) a)])]/2 && a != 0) || (x == -b/(4 a) + Sqrt[b^2/(4 a^2) - (2 c)/(3 a) + (2^(1/3) (c^2 - 3 b d + 12 a e))/(3 a (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)) + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)/(3 2^(1/3) a)]/2 + Sqrt[b^2/(2 a^2) - (4 c)/(3 a) - (2^(1/3) (c^2 - 3 b d + 12 a e))/(3 a (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)) - (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e)^2])^(1/3)/(3 2^(1/3) a) + (-(b^3/a^3) + (4 b c)/a^2 - (8 d)/a)/(4 Sqrt[b^2/(4 a^2) - (2 c)/(3 a) + (2^(1/3) (c^2 - 3 b d + 12 a e))/(3 a (2 c^3 - 9 b c d + 27 a d^2 + 27 b^2 e - 72 a c e + Sqrt[-4 (c^2 - 3 b d + 12 a e)^3 + (2 c^3 - 9 b c d + 27 a

In what way would this ever be useful, or even usable, on a calculator? There are better methods for getting roots once you move beyond basic polynomials.