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)#