Question

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

Answer 1

Use Cardano's method to find Real zero:

`x_1 = 1/27(1+root(3)((6235+27sqrt(46221))/2)+root(3)((6235-27sqrt(46221))/2))`

and related Complex zeros.

Explanation:

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`f(x) = -9x^3+x^2+4x-3`

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

`Delta = 16+2304+12-19683+1944 = -15407`

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=-2187f(x)=19683x^3-2187x^2-8748x+6561`

`=(27x-1)^3-327(27x-1)+6235`

`=t^3-327t+6235`

where `t=(27x-1)`

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Cardano's method

We want to solve:

`t^3-327t+6235=0`

Let `t=u+v`.

Then:

`u^3+v^3+3(uv-109)(u+v)+6235=0`

Add the constraint `v=109/u` to eliminate the `(u+v)` term and get:

`u^3+1295029/u^3+6235=0`

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

`(u^3)^2+6235(u^3)+1295029=0`

Use the quadratic formula to find:

`u^3=(-6235+-sqrt((6235)^2-4(1)(1295029)))/(2*1)`

`=(6235+-sqrt(38875225-5180116))/2`

`=(6235+-sqrt(33695109))/2`

`=(6235+-27sqrt(46221))/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)((6235+27sqrt(46221))/2)+root(3)((6235-27sqrt(46221))/2)`

and related Complex roots:

`t_2=omega root(3)((6235+27sqrt(46221))/2)+omega^2 root(3)((6235-27sqrt(46221))/2)`

`t_3=omega^2 root(3)((6235+27sqrt(46221))/2)+omega root(3)((6235-27sqrt(46221))/2)`

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

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

`x_1 = 1/27(1+root(3)((6235+27sqrt(46221))/2)+root(3)((6235-27sqrt(46221))/2))`

`x_2 = 1/27(1+omega root(3)((6235+27sqrt(46221))/2)+omega^2 root(3)((6235-27sqrt(46221))/2))`

`x_3 = 1/27(1+omega^2 root(3)((6235+27sqrt(46221))/2)+omega root(3)((6235-27sqrt(46221))/2))`