Kirill Z. answered • 02/21/14

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In this case, you can notice that your polynomial can be factored as follows:

6x

^{4}-4x^{3}-3x+2=(6x^{4}-3x)-(4x^{3}-2)=3x(2x^{3}-1)-2(2x^{3}-1)=(3x-2)(2x^{3}-1)The roots of your polynomial are either the roots of the first term, or the second. The root of the first term is 2/3. The second term boils down to:

2x

^{3}=1 or x=(1/2)^{1/3}Obviously, cube root of 1/2 is irrational (two other roots are complex), thus the only rational root in this case is x=2/3