Equations of 4-th Degree
The program determines the real solutions of an equation of the 4th or of a lower degree.
a·x4 + b·x3 + c·x2 + d·x + e = 0
Example:
To determine the solutions of the equation x4 + 2x3 - 8x2 -18x - 9 = 0 , enter the coefficients a to e as follows:
and receive the set of solutions:
x4 + 2·x3 - 8·x2 - 18·x - 9 = 0 <=> (x + 1)2·(x - 3)·(x + 3) = 0
L = {-3; -1; 3}
The formula for quadratic equations is well known. The formula for cubic equations was derived by Scipione del Ferro in 1530, but published after his death by his adept Ceralamo Cardano . The extension to equations of 4-th degree Cardano himself attributed to his adept Lodovico Ferrari .
Equations of 5th or higher degree
We have to thank the Norwegian mathematician Niels Henrik Abel for the proof that no formula can exist for equations with higher degree than 4. The only solutions are approximations like we do in Curve Sketching.
If we know one solution of an equation, we sometimes can find the others by polynomial division.
For example x5 - 12x3 - 2x2 + 27x + 18 = 0 has the solution x1 = 2 .
Thus we know that the left site of the equation can be divided by (x - 2) without a remainder.
(x5 - 12x3 - 2x2 + 27x + 18) : (x - 2 ) = x4 + 2x3- 8x2 - 18x - 9
The equation x4 + 2x3- 8x2 - 18x - 9 = 0 allows us to find the missing solutions.
Factorization of polynomials is even faster. It provides all rational solutions:
p(x) = x5 - 12x3 - 2x2 + 27x + 18
= (x + 1)2·(x - 2)·(x - 3)·(x + 3)
Rational zeros: -1, 2, 3, -3