Question

How do you find all the zeros of `f(x)= 2x^3 - 6x^2 + 7x +9`?

Answer 1

Use Cardano's method to find Real zero:

`x_1 = 1/6(6+root(3)(648+6sqrt(11670))+root(3)(648-6sqrt(11670)))`

and related Complex zeros.

Explanation:

`f(x) = 2x^3-6x^2+7x+9`

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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=-6`, `c=7` and `d=9`, so we find:

`Delta = 1764-2744+7776-8748-13608 = -15560`

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=4f(x)=8x^3-24x^2+28x+36`

`=(2x-2)^3+2(2x-2)+48`

`=t^3+2t+48`

where `t=(2x-2)`

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Cardano's method

We want to solve:

`t^3+2t+48=0`

Let `t=u+v`.

Then:

`u^3+v^3+(3uv+2)(u+v)+48=0`

Add the constraint `v=-2/(3u)` to eliminate the `(u+v)` term and get:

`u^3-8/(27u^3)+48=0`

Multiply through by `27u^3` and rearrange slightly to get:

`27(u^3)^2+1296(u^3)-8=0`

Use the quadratic formula to find:

`u^3=(-1296+-sqrt((1296)^2-4(27)(-8)))/(2*27)`

`=(1296+-sqrt(1679616+864))/54`

`=(1296+-sqrt(1680480))/54`

`=(1296+-12sqrt(11670))/54`

`=(648+-6(11670))/27`

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=1/3(root(3)(648+6sqrt(11670))+root(3)(648-6sqrt(11670)))`

and related Complex roots:

`t_2=1/3(omega root(3)(648+6sqrt(11670))+omega^2 root(3)(648-6sqrt(11670)))`

`t_3=1/3(omega^2 root(3)(648+6sqrt(11670))+omega root(3)(648-6sqrt(11670)))`

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

Now `x=1/2(2+t)=1/6(6+3t)`. So the roots of our original cubic are:

`x_1 = 1/6(6+root(3)(648+6sqrt(11670))+root(3)(648-6sqrt(11670)))`

`x_2 = 1/6(6+omega root(3)(648+6sqrt(11670))+omega^2 root(3)(648-6sqrt(11670)))`

`x_3 = 1/6(6+omega^2 root(3)(648+6sqrt(11670))+omega root(3)(648-6sqrt(11670)))`