# How do you find all the zeros of #f(x)=x^5+3x^3-x+6#?

##### 1 Answer

Find there are no rational zeros.

Use Durand-Kerner or similar to find approximations.

#### Explanation:

By the rational root theorem any *rational* zeros of

That means that the only possible rational zeros are:

#+-1# ,#+-2# ,#+-3# ,#+-6#

None of these work, so

In common with most quintic polynomials, the zeros are not expressible in terms of

#x ~~ -1.17826#

#x ~~ -0.202554+-1.89313i#

#x ~~ 0.791684+-0.882051i#

See https://socratic.org/s/avxUUEiJ for another example.

Here's a sample C++ program that implements the Durand-Kerner algorithm for this example: