# How do you use the rational roots theorem to find all possible zeros of #f(x) = 6x^4 + 2x^3 -3x^2 +2#?

##### 1 Answer

This quartic has no rational zeros. It has

#x_(1,2) ~~ 0.56707+-0.43127i#

#x_(3,4) ~~ -0.73373+-0.34406i#

#### Explanation:

#f(x) = 6x^4+2x^3-3x^2+2#

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

So the only possible *rational* zeros of

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

Trying any of these, we find *rational* zeros.

In fact, this quartic has only Complex zeros, which it is possible, but horribly messy to find algebraically. We can use numerical methods such as Newton's method or Durand-Kerner to find approximations:

#x_(1,2) ~~ 0.56707+-0.43127i#

#x_(3,4) ~~ -0.73373+-0.34406i#