Question

How do you find all the zeros of `F(X)= 3x^2 - 2x^2 + x +4`?

Answer 1

Use Cardano's method to find Real zero:

`x_1 = 1/9(2+root(3)(505+135sqrt(14))+root(3)(505-135sqrt(14)))`

and related Complex zeros...

Explanation:

Assuming that there's a typo in the question...

`f(x) = 3x^3-2x^2+x+4`

`color(white)()`
Descriminant

The discriminant `Delta` of a cubic polynomial in the form `ax^3+bx^2+cx+d` is given by the formula:

`Delta = b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd`

In our example, `a=3`, `b=-2`, `c=1` and `d=4`, so we find:

`Delta = 4-12+128-3888-432 = -4200`

Since `Delta < 0` this cubic has `1` Real zero and `2` non-Real Complex zeros, which are Complex conjugates of one another.

`color(white)()`
Tschirnhaus transformation

To make the task of solving the cubic simpler, we make the cubic simpler using a linear substitution known as a Tschirnhaus transformation.

`0=243f(x)=729x^3-486x^2+243x+972`

`=(9x-2)^3+15(9x-2)+1010`

`=t^3+15t+1010`

where `t=(9x-2)`

`color(white)()`
Cardano's method

We want to solve:

`t^3+15t+1010=0`

Let `t=u+v`.

Then:

`u^3+v^3+3(uv+5)(u+v)+1010=0`

Add the constraint `v=-5/u` to eliminate the `(u+v)` term and get:

`u^3-125/u^3+1010=0`

Multiply through by `u^3` and rearrange slightly to get:

`(u^3)^2+1010(u^3)-125=0`

Use the quadratic formula to find:

`u^3=(-1010+-sqrt((1010)^2-4(1)(-125)))/(2*1)`

`=(1010+-sqrt(1020100+500))/2`

`=(1010+-sqrt(1020600))/2`

`=(1010+-270sqrt(14))/2`

`=505+-135sqrt(14)`

Since this is Real and the derivation is symmetric in `u` and `v`, we can use one of these roots for `u^3` and the other for `v^3` to find Real root:

`t_1=root(3)(505+135sqrt(14))+root(3)(505-135sqrt(14))`

and related Complex roots:

`t_2=omega root(3)(505+135sqrt(14))+omega^2 root(3)(505-135sqrt(14))`

`t_3=omega^2 root(3)(505+135sqrt(14))+omega root(3)(505-135sqrt(14))`

where `omega=-1/2+sqrt(3)/2i` is the primitive Complex cube root of `1`.

Now `x=1/9(2+t)`. So the zeros of our original cubic are:

`x_1 = 1/9(2+root(3)(505+135sqrt(14))+root(3)(505-135sqrt(14)))`

`x_2 = 1/9(2+omega root(3)(505+135sqrt(14))+omega^2 root(3)(505-135sqrt(14)))`

`x_3 = 1/9(2+omega^2 root(3)(505+135sqrt(14))+omega root(3)(505-135sqrt(14)))`