How do you find all rational roots for x3−3x2+4x−12=0?

1 Answer
Shwetank Mauria
Apr 16, 2016

The only rational root of x3−3x2+4x−12=0 is 3.

Explanation:

x3−3x2+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 x3−3x2+4x−12. Dividing latter by (x−3), we get

x3−3x2+4x−12=x2(x−3)+4(x−3)=(x2+4)(x−3)

x2+4=0 does not have rational rots as discriminant b2−4ac=0−4⋅1⋅4=−16

hence the only rational root of x3−3x2+4x−12=0 is 3.

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

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