Question

How do you find all the real and complex roots of `f (x) = x^5 – x^4 + 3x^3 + 9x^2 – x + 5`?

Answer 1

Use the Durand-Kerner method to find numerical approximations for the zeros.

Explanation:

Given:

`f(x) = x^5-x^4+3x^3+9x^2-x+5`

By the rational root 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 `5` and `q` a divisor of the coefficient `1` of the leading term.

So the only possible rational zeros are:

`+-1`, `+-5`

None of these work, so the zeros of `f(x)` are irrational and/or Complex.

To find numeric approximations for the zeros, you can use the Durand-Kerner method:

Suppose the `5` zeros are: `p, q, r, s, t`.

Choose initial approximations for these zeros as follows:

`p_0 = (0.4+0.9i)^0`

`q_0 = (0.4+0.9i)^1`

`r_0 = (0.4+0.9i)^2`

`s_0 = (0.4+0.9i)^3`

`t_0 = (0.4+0.9i)^4`

Then iterate using the formulas:

`p_(i+1) = p_i-(f(p_i))/((p_i-q_i)(p_i-r_i)(p_i-s_i)(p_i-t_i))`

`q_(i+1) = q_i-(f(q_i))/((q_i-p_(i+1))(q_i-r_i)(q_i-s_i)(q_i-t_i))`

`r_(i+1) = r_i-(f(r_i))/((r_i-p_(i+1))(r_i-q_(i+1))(r_i-s_i)(r_i-t_i))`

`s_(i+1) = s_i-(f(s_i))/((s_i-p_(i+1))(s_i-q_(i+1))(s_i-r_(i+1))(s_i-t_i))`

`t_(i+1) = t_i-(f(t_i))/((t_i-p_(i+1))(t_i-q_(i+1))(t_i-r_(i+1))(t_i-s_(i+1))`

Keep iterating until the values are stable to the desired accuracy.

The zeros you will find will be approximately:

`-1.6073`

`0.109713+0.702004i`

`0.109713-0.702004i`

`1.19394+2.17633i`

`1.19394-2.17633i`

Here's a C++ program which does the calculation:

This is what its (slightly primitive) output looks like: