Question

How do you find all the zeros of `x^3-4x^2+3x-1`?

Answer 1

Use Cardano's method to find Real zero:

`1/3(4+root(3)((-47+3sqrt(93))/2)+root(3)((-47-3sqrt(93))/2))`

and Complex conjugate pair of related zeros.

Explanation:

`f(x) = x^3-4x^2+3x-1`

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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=1`, `b=-4`, `c=3` and `d=-1`, so we find:

`Delta = 144-108-256-27+216 = -31`

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=27f(x)=27x^3-108x^2+81x-27`

`=(3x-4)^3-21(3x-4)-47`

`=t^3-21t-47`

where `t=(3x-4)`

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Cardano's method

We want to solve:

`t^3-21t-47=0`

Let `t=u+v`.

Then:

`u^3+v^3+3(uv-7)(u+v)-47=0`

Add the constraint `v=7/u` to eliminate the `(u+v)` term and get:

`u^3+343/u^3-47=0`

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

`(u^3)^2-47(u^3)+343=0`

Use the quadratic formula to find:

`u^3=(47+-sqrt((-47)^2-4(1)(343)))/(2*1)`

`=(-47+-sqrt(2209-1372))/2`

`=(-47+-sqrt(837))/2`

`=(-47+-3sqrt(93))/2`

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)((-47+3sqrt(93))/2)+root(3)((-47-3sqrt(93))/2)`

and related Complex roots:

`t_2=omega root(3)((-47+3sqrt(93))/2)+omega^2 root(3)((-47-3sqrt(93))/2)`

`t_3=omega^2 root(3)((-47+3sqrt(93))/2)+omega root(3)((-47-3sqrt(93))/2)`

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

Now `x=1/3(4+t)`. So the roots of our original cubic are:

`x_1 = 1/3(4+root(3)((-47+3sqrt(93))/2)+root(3)((-47-3sqrt(93))/2))`

`x_2 = 1/3(4+omega root(3)((-47+3sqrt(93))/2)+omega^2 root(3)((-47-3sqrt(93))/2))`

`x_3 = 1/3(4+omega^2 root(3)((-47+3sqrt(93))/2)+omega root(3)((-47-3sqrt(93))/2))`