Question

How do you find all the zeros of `g(x) = 9x^3- 7x^2 + 10x - 4`?

Answer 1

Use Cardano's method to find Real zero:

`x_1 = 1/27(7+root(3)(-1882+81sqrt(2185))+root(3)(-1882-81sqrt(2185)))`

and related Complex zeros.

Explanation:

`g(x) = 9x^3-7x^2+10x-4`

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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=9`, `b=-7`, `c=10` and `d=-4`, so we find:

`Delta = 4900-36000-5488-34992+45360 = -26220`

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=2187g(x)=19683x^3-15309x^2+21870x-8748`

`=(27x-7)^3+663(27x-7)-3764`

`=t^3+663t-3764`

where `t=(27x-7)`

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Cardano's method

We want to solve:

`t^3+663t-3764=0`

Let `t=u+v`.

Then:

`u^3+v^3+3(uv+221)(u+v)-3764=0`

Add the constraint `v=-221/u` to eliminate the `(u+v)` term and get:

`u^3-10793861/u^3-3764=0`

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

`(u^3)^2-3764(u^3)-10793861=0`

Use the quadratic formula to find:

`u^3=(3764+-sqrt((-3764)^2-4(1)(-10793861)))/(2*1)`

`=(-3764+-sqrt(14167696+43175444))/2`

`=(-3764+-sqrt(57343140))/2`

`=(-3764+-162(2185))/2`

`=-1882+-81(2185)`

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)(-1882+81sqrt(2185))+root(3)(-1882-81sqrt(2185))`

and related Complex roots:

`t_2=omega root(3)(-1882+81sqrt(2185))+omega^2 root(3)(-1882-81sqrt(2185))`

`t_3=omega^2 root(3)(-1882+81sqrt(2185))+omega root(3)(-1882-81sqrt(2185))`

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

Now `x=1/27(7+t)`. So the zeros of our original cubic are:

`x_1 = 1/27(7+root(3)(-1882+81sqrt(2185))+root(3)(-1882-81sqrt(2185)))`

`x_2 = 1/27(7+omega root(3)(-1882+81sqrt(2185))+omega^2 root(3)(-1882-81sqrt(2185)))`

`x_3 = 1/27(7+omega^2 root(3)(-1882+81sqrt(2185))+omega root(3)(-1882-81sqrt(2185)))`