# How do you use the rational roots theorem to find all possible zeros of #f(x)=x^3-5x^2+2x+12#?

##### 1 Answer

Jul 24, 2016

#### Explanation:

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

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

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

Trying each in turn we find:

#f(3) = 27-45+6+12 = 0#

So

#x^3-5x^2+2x+12#

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

#= (x-3)(x^2-2x+1-5)#

#= (x-3)((x-1)^2-(sqrt(5))^2)#

#= (x-3)(x-1-sqrt(5))(x-1+sqrt(5))#

So the remaining two zeros are:

#x = 1+-sqrt(5)#