Question

How do you find all the zeros of `f(x)=x^3+x^2-11x-30`?

Answer 1

Use Cardano's method to find Real zero:

`x_1 = 1/3(-1+root(3)((-709+9sqrt(4265))/2)+root(3)((-709-9sqrt(4265))/2))`

and related Complex zeros.

Explanation:

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`f(x) = x^3+x^2-11x-30`

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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=1`, `c=-11` and `d=-30`, so we find:

`Delta = 121+5324+120-24300+5940 = -12795`

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+27x^2-297x-810`

`=(3x+1)^3-102(3x+1)-709`

`=t^3-102t-709`

where `t=(3x+1)`

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Cardano's method

We want to solve:

`t^3-102t-709=0`

Let `t=u+v`.

Then:

`u^3+v^3+3(uv-34)(u+v)-709=0`

Add the constraint `v=34/u` to eliminate the `(u+v)` term and get:

`u^3+39304/u^3-709=0`

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

`(u^3)^2-709(u^3)+39304=0`

Use the quadratic formula to find:

`u^3=(709+-sqrt((-709)^2-4(1)(39304)))/(2*1)`

`=(-709+-sqrt(502681-157216))/2`

`=(-709+-sqrt(345465))/2`

`=(-709+-9sqrt(4265))/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)((-709+9sqrt(4265))/2)+root(3)((-709-9sqrt(4265))/2)`

and related Complex roots:

`t_2=omega root(3)((-709+9sqrt(4265))/2)+omega^2 root(3)((-709-9sqrt(4265))/2)`

`t_3=omega^2 root(3)((-709+9sqrt(4265))/2)+omega root(3)((-709-9sqrt(4265))/2)`

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

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

`x_1 = 1/3(-1+root(3)((-709+9sqrt(4265))/2)+root(3)((-709-9sqrt(4265))/2))`

`x_2 = 1/3(-1+omega root(3)((-709+9sqrt(4265))/2)+omega^2 root(3)((-709-9sqrt(4265))/2))`

`x_3 = 1/3(-1+omega^2 root(3)((-709+9sqrt(4265))/2)+omega root(3)((-709-9sqrt(4265))/2))`