The old mathematicians(like Euclid: around 330 bc - around 275 bc) ran into the problem of solving a polynomial of degree 3 a very long time ago, when trying to find the solution for various geometrical challenges...(like the doubling of the volume of a cube, trisection of an angle..).All these problems cannot be solved by instruments like compasses or a ruler. There are clear indications that finding solutions for this particular problem even goes back to ancient Babylonia (approx. 2'000 bc). The Arabian mathematicians inspired the famous Italian mathematician Leonardo Pisano Fibonacci(1170 - 1250), who found a solution for a special cubic equation (where c

The general case(except from the pardonable coefficient 1 with x

The solution of a cubic polynomial is linked with the Renaissance mathematician Girolamo Cardano: 24.9.1501(Pavia,Italy) - 21.9.1576(Rome,Italy). But now lets look at our polynomial P(x) above (assuming only REAL coefficients). By the substitution z = x - c

u

But this way we obtain 3 solutions each for u and v, leaving us with 9 solutions in total for our x = u + v at first glance. But we remember that 3uv = - p, eg once we choose one solution for u, v is determined, which reduces the total number of solutions for u + v from 9 to 3. In todays terms, we can say that these 3 solutions must have the form x

Now lets look at one example, an example our Renaissance mathematicians could have solved for us:

x

The transform of x = r + 1 leads us to : r

The

(a) we continue with the negative D and use complex numbers ..

(b) we use the equivalence of cos 3α = 4cos

But the period of cosine is 2π so if we set: 3α = φ + 2πn we will get: z

f:{(signum x)*(abs x) xexp 1%3};s3:sqrt 3;pi:acos -1f

s:s%first s;p:(b:s@2)-a*t:(a:s@1)%3;c:-0.5*q:(s@3)-t*b-2f*t*t;D:(c*c)+p*p*p%27f;

/cardano,ferro,tartaglia

if[D>=0f;u:f c+w:sqrt D;v:f c-w;mn0:neg m0:0.5*m:u+v;n0:s3*0.5*n:u-v;r:(m,0f;mn0,-1f*n0;mn0,n0);r[;0]:r[;0]-t]

/casus irreducibilis; using (b)

if[D<0f;h:sqrt -4f*p%3;i:acos -4f*q%h*h*h;r:(h*cos (i+pi*0 2 4)%3)-t]

s:s%first s;p:(b:s@2)-a*t:(a:s@1)%3;c:-0.5*q:(s@3)-t*b-2f*t*t;D:(c*c)+p*p*p%27f;

/cardano,ferro,tartaglia

if[D>=0f;u:f c+w:sqrt D;v:f c-w;mn0:neg m0:0.5*m:u+v;n0:s3*0.5*n:u-v;r:(m,0f;mn0,-1f*n0;mn0,n0);r[;0]:r[;0]-t]

/casus irreducibilis; using (b)

if[D<0f;h:sqrt -4f*p%3;i:acos -4f*q%h*h*h;r:(h*cos (i+pi*0 2 4)%3)-t]

Sample 1: x

Sample 2: x

Sample 3: x

Sample 4: 144x

Sample 5: z

this can be reduced to (a-b)(a+b)(a

In other words: z

©++ MILAN ONDRUŠ