How do you find all rational roots for #x^3 - 3x^2 + 4x - 12 = 0#?

1 Answer
Apr 16, 2016

Answer:

The only rational root of #x^3-3x^2+4x-12=0# is #3#.

Explanation:

#x^3-3x^2+4x-12=0# can have one root among factors of #12# i.e. #{1,-1,2,-2,3,-3,4,-4,6,-6,12,-12}#, if at least one root is rational.

It is apparent that #3# satisfies the equation, hence #x-3# is a factor of #x^3-3x^2+4x-12#. Dividing latter by #(x-3)#, we get

#x^3-3x^2+4x-12=x^2(x-3)+4(x-3)=(x^2+4)(x-3)#

#x^2+4=0# does not have rational rots as discriminant #b^2-4ac=0-4*1*4=-16#

hence the only rational root of #x^3-3x^2+4x-12=0# is #3#.

The two roots will be imaginary numbers #-2i# and #+2i#.