All rational roots that are possible according to the rational zero theorem? ##x^3-34x+12=0##

According to the theorem, the possible rational roots are:

##+-1##, ##+-2##, ##+-3##, ##+-4##, ##+-6##, ##+-12##

##f(x) = x^3-34x+12##

By the rational root theorem, any rational zeros of ##f(x)## are expressible in the form ##p/q## for integeres ##p, q## with ##p## a divisor of the constant term ##12## and ##q## a divisor of the coefficient ##1## of the leading term.

That means that the only possible rational zeros are:

##+-1##, ##+-2##, ##+-3##, ##+-4##, ##+-6##, ##+-12##

Trying each in turn, we eventually find that:

##f(color(blue)(-6)) = (color(blue)(-6))^3-34(color(blue)(-6))+12##

##color(white)(f(color(white)(-6))) = -216+204+12##

##color(white)(f(color(white)(-6))) = 0##

So ##x=-6## is a rational root.

The other two roots are Real but irrational.