# How do you use the rational roots theorem to find all possible zeros of #f(x)=x^5-3x^2-4#?

##### 1 Answer

Find

We can find numerical approximations:

#x_1 ~~ 1.6477#

#x_(2,3) ~~ 0.151952+-1.03723i#

#x_(4,5) ~~ -0.975801+-1.12111i#

#### Explanation:

#f(x) = x^5-3x^2-4#

By the rational roots theorem, any *rational* zeros of

That means that the only possible *rational* zeros are:

#+-1, +-2, +-4#

None of these work, so

In common with most quitics and polynomials of higher degree, the zeros of this one cannot be expressed in terms of

It is possible to find numerical approximations for the zeros using a method like Durand-Kerner.

For example, we can use this C++ program...

to find numerical approximations for the zeros:

#x_1 ~~ 1.6477#

#x_(2,3) ~~ 0.151952+-1.03723i#

#x_(4,5) ~~ -0.975801+-1.12111i#

For a little more explanation of the method see https://socratic.org/s/ax2iiWhR