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

#f(x) = 3x^5-2x^2+16x-2#

By the rational roots 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 #-2# and #q# a divisor of the coefficient #3# of the leading term.

That means that the only possible rational zeros of #f(x)# are:

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

You can substitute each of these into the formula for #f(x)# to find that none of them is a zero.

So this quintic has no rational zeros.

#color(white)()#
By the fundamental theorem of algebra it does have exactly #5# Complex, possibly Real, zeros, counting multiplicity.

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 #n#th roots.

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 #5# zeros at once.

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: