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 get the solution set:

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. 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  x⁵ - 12x³ - 2x² + 27x + 18 = 0  has the solution  x₁ = 2 .

Thus we know that the left side 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 gives 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

See also:

Wikipedia:Cubic function
Wikipedia: Scipione del Ferro | Gerolamo Cardano | Ludovico Ferrari