Question

How do you find the zeros of `f(x) = 2x^3+2x^2-2x-1` ?

Answer 1

Use a trigonometric substitution method to find zeros:

`x_n = 1/3(4 cos(1/3 cos^(-1)(5/32) + (2npi)/3) - 1)" "`

for `n = 0, 1, 2`.

Explanation:

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

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Rational roots theorem

By the rational roots theorem, any rational zeros of `f(x)` are expressible in the form `p/q` for integers `p, q` with `p` a divisor of the constant term `-1` and `q` a divisor of the coefficient `2` of the leading term.

That means that the only possible rational zeros are:

`+-1/2`, `+-1`

None of these work, so `f(x)` has no rational zeros.

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Discriminant

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=2`, `c=-2` and `d=-1`, so we find:

`Delta = 16+64+32-108+144 = 148`

Since `Delta > 0` this cubic has `3` Real zeros.

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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=108f(x)=216x^3+216x^2-216x-108`

`=(6x+2)^3-48(6x+2)-20`

`=t^3-48t-20`

where `t=(6x+2)`

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Trigonometric substitution

Look for a substitution of the form `t = k cos theta` with `k` chosen so that the resulting cubic contains:

`4cos^3 theta - 3 cos theta = cos 3 theta`

Let `t = k cos theta`

`0 = t^3-48t-20`

`color(white)(0) = k^3 cos^3 theta - 48k cos theta - 20`

`color(white)(0) = k (k^2 cos^3 theta - 48 cos theta) - 20`

Let `k=8`

`0 = 8 (64 cos^3 theta - 48 cos theta) - 20`

`color(white)(0) = 128 (4 cos^3 theta - 3 cos theta) - 20`

`color(white)(0) = 128 cos 3 theta - 20`

So:

`cos 3 theta = 20/128 = 5/32`

Hence:

`3 theta = +-cos^(-1)(5/32) + 2npi`

So:

`theta = +-1/3 cos^(-1)(5/32) + (2npi)/3`

Note that `cos (-theta) = cos (theta)`, so we can discard the `+-` to get distinct solutions:

`t_n = 8 cos (1/3 cos^(-1)(5/32) + (2npi)/3)" "` for `n = 0, 1, 2`

Then `x = 1/6(t-2)`

`x_n = 1/3(4 cos(1/3 cos^(-1)(5/32) + (2npi)/3) - 1)" "` for `n = 0, 1, 2`