## Equations of 4-th Degree

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

### Example:

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 get the solution set:

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 * .

### 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. What then remains are approximate calculations such as those used to determine the zeros in the program section
*Calculus of arbitrairy Functions*.

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 side 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 gives 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:Cubic functionWikipedia: Scipione del Ferro | Gerolamo Cardano | Ludovico Ferrari