# How do you find all the real and complex roots of x^5 + 7*x^3 + 3*x + 2 = 0?

Jun 4, 2016

Use a numerical method to find approximations for the roots:

$x \approx - 0.44915$

$x \approx 0.249655 \pm 0.786034 i$

$x \approx - 0.0250794 \pm 2.55851 i$

#### Explanation:

$f \left(x\right) = {x}^{5} + 7 {x}^{3} + 3 x + 2$

In common with quintics in general, this quintic has no zeros expressible in algebraic terms using $n$th roots.

We can find numerical approximations for the zeros using a method such as Durand-Kerner.

For another quintic solution approximated using Durand-Kerner, see https://socratic.org/s/av3SsZ5D

In the current example we find approximations:

$x \approx - 0.44915$

$x \approx 0.249655 \pm 0.786034 i$

$x \approx - 0.0250794 \pm 2.55851 i$

I used this C++ program: