The program determines the real solutions of an equation of the 4th or of a lower degree.

a·x^{4} + b·x^{3} + c·x^{2} + d·x + e = 0

To determine the solutions of the equation x^{4} + 2x^{3} - 8x^{2} -18x - 9 = 0 ,
enter the coefficients a to e as follows:

and receive the set of solutions:

x^{4}+ 2·x^{3}- 8·x^{2}- 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 * .

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 x^{5} - 12x^{3} - 2x^{2} + 27x + 18 = 0
has the solution x_{1} = 2 .

Thus we know that the left site of the equation can be divided by (x - 2) without a remainder.

(x^{5} - 12x^{3} - 2x^{2} + 27x + 18) : (x - 2 ) = x^{4} + 2x^{3}- 8x^{2} - 18x - 9

The equation x^{4} + 2x^{3}- 8x^{2} - 18x - 9 = 0 allows us to find the missing solutions.

Factorization of polynomials is even faster. It provides all rational solutions:

p(x) = x^{5}- 12x^{3}- 2x^{2}+ 27x + 18 = (x + 1)^{2}·(x - 2)·(x - 3)·(x + 3) Rational zeros: -1, 2, 3, -3

See also:

Wikipedia: Scipione del Ferro | Gerolamo Cardano | Ludovico Ferrari