code:But, how does one solve a "cubatic" equasion? I tried simplifying:ax^2 + bx + c = 0 /4a
4a^2x^2 + 4abx + 4ac = 0 /- 4ac + b^2
4a^2x^a + 4abx + b^2 = b^2 - 4ac
(2ax + b)^2 = b^2 - 4ac /SQRT()
2ax + b = +/- SQRT(b^2 - 4ac) /- b
2ax = - b +/- SQRT(b^2 - 4ac) /: 2a
x = [-b +/- SQRT(b^2 - 4ac)]:2a
code:But... If I get rid of the d, it does not help me, because after dividing by x, it's d:x on the other side. As for dividing by D and getting rid of the "1", I can't say that when x(ax^2 + bx + c) = -1 either x or ax^2 + bx + c equals to -1, because it only works like that with zero or infinity.ax^3 + bx^2 + cx + d = 0
x(ax^2 + bx + c) + d = 0
quote:See http://mathworld.wolfram.com/Polynomial.html
Polynomials of orders one to four are solvable using only rational operations and finite root extractions. A first-order equation is trivially solvable. A second-order equation is soluble using the quadratic equation. A third-order equation is solvable using the cubic equation. A fourth-order equation is solvable using the quartic equation. It was proved by Abel Eric Weisstein's World of Biography and Galois Eric Weisstein's World of Biography using group theory that general equations of fifth and higher order cannot be solved rationally with finite root extractions (Abel's impossibility theorem).
However, the general quintic equation may be given in terms of the Jacobi theta functions, or hypergeometric functions in one variable. Hermite Eric Weisstein's World of Biography and Kronecker Eric Weisstein's World of Biography proved that higher order polynomials are not soluble in the same manner. Klein showed that the work of Hermite was implicit in the group properties of the icosahedron. Klein's method of solving the quintic in terms of hypergeometric functions in one variable can be extended to the sextic, but for higher order polynomials, either hypergeometric functions in several variables or "Siegel functions" must be used (Belardinelli 1960, King 1996, Chow 1999). In the 1880s, Poincaré Eric Weisstein's World of Biography created functions which give the solution to the nth order polynomial equation in finite form. These functions turned out to be "natural" generalizations of the elliptic functions.
quote:There's kind of a combination method that is not so impossible. For a polynomial equation y = Ax^n + . . . . + B, you can, first of all, use desCartes's Rule of Signs to get a sense of how many positive and how many negative roots there may be. Then, you can make a list of all the possible rational roots, because all the possible rational roots must be of the form
A polynomial equation may be solved either numerically (trying different values until an intercept is found - there's a method for selecting the next guess that makes this less difficult than it sounds) or by factoring.
The first method is really only suitable for computers. . . .
code:(I'm too lazy to explain why now, but if you can't figure it out and are itching to know, I'll be happy to do so.)
a factor of B
root = ---------------
a factor of A
code:We'll use 0 as our guess.f(x) = 1 - x + 0.5 * x ^ 3
f'(x) = -1 + 3/2 * x ^ 2
newtons's method: xn+1 = xn - f(xn) / f'(xn)
code:By using bisection first we can get by most of these kind of problems.x0 = 0;
x1 = 1;
x2 = 0;
x3 = 1;
x4 = 0;
...