# How do you use the rational roots theorem to find all possible zeros of #f(x) = 3X^5 - 2X^2 + 16X - 2#?

##### 1 Answer

Use a numerical method to find:

#x ~~ 0.12701#

#x ~~ 1.03966+-0.998855i#

#x ~~ -1.10317+-1.14378i#

#### Explanation:

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

That means that the only possible *rational* zeros of

#+-1/3# ,#+-2/3# ,#+-1# ,#+-2#

You can substitute each of these into the formula for

So this quintic has no *rational* zeros.

By the fundamental theorem of algebra it does have exactly

In addition, since its coefficients are Real, any non-Real Complex zeros must occur in Complex conjugate pairs.

Typically for a quintic, the zeros of this particular one are not expressible in terms of

About the best we can do is find numeric approximations using a method like Durand-Kerner. See https://socratic.org/s/avyDEG5X for a fuller explanation, but the Durand-Kerner algorithm finds approximations for all

The results I found were:

#x ~~ 0.12701#

#x ~~ 1.03966+-0.998855i#

#x ~~ -1.10317+-1.14378i#

Here's the C++ program I used to find them: