Question

How do you find all the zeros of `g(x)= - 2x^3+5x^2-6x-10`?

Answer 1

Use Cardano's method to find Real zero:

`x_1 = 1/6(5+root(3)(685+6sqrt(13071))+root(3)(685-6sqrt(13071)))`

and related Complex zeros.

Explanation:

`g(x) = -2x^3+5x^2-6x-10`

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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=-2`, `b=5`, `c=-6` and `d=-10`, so we find:

`Delta = 900-1728+5000-10800-10800 = -17428`

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

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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=-108g(x)=216x^3-540x^2+648x+1080`

`=(6x-5)^3+33(6x-5)+1370`

`=t^3+33t+1370`

where `t=(6x-5)`

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Cardano's method

We want to solve:

`t^3+33t+1370=0`

Let `t=u+v`.

Then:

`u^3+v^3+3(uv+11)(u+v)+1370=0`

Add the constraint `v=-11/u` to eliminate the `(u+v)` term and get:

`u^3-1331/u^3+1370=0`

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

`(u^3)^2+1370(u^3)-1331=0`

Use the quadratic formula to find:

`u^3=(-1370+-sqrt((1370)^2-4(1)(-1331)))/(2*1)`

`=(1370+-sqrt(1876900+5324))/2`

`=(1370+-sqrt(1882224))/2`

`=(1370+-12sqrt(13071))/2`

`=685+-6sqrt(13071)`

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)(685+6sqrt(13071))+root(3)(685-6sqrt(13071))`

and related Complex roots:

`t_2=omega root(3)(685+6sqrt(13071))+omega^2 root(3)(685-6sqrt(13071))`

`t_3=omega^2 root(3)(685+6sqrt(13071))+omega root(3)(685-6sqrt(13071))`

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

Now `x=1/6(5+t)`. So the zeros of our original cubic are:

`x_1 = 1/6(5+root(3)(685+6sqrt(13071))+root(3)(685-6sqrt(13071)))`

`x_2 = 1/6(5+omega root(3)(685+6sqrt(13071))+omega^2 root(3)(685-6sqrt(13071)))`

`x_3 = 1/6(5+omega^2 root(3)(685+6sqrt(13071))+omega root(3)(685-6sqrt(13071)))`