list all the possible rational roots of P(x) given by the Rational Root Theorom.
P(x)=4x^4-2x^3+x^2-12
the numerator must be a factor of 12
The denominator must be a factor of 4
So the possibilities are
±(12/1), ±(12/2), ±(12/4)
±(6/1), ±(6/2), ±(6/4)
±(4/1), ±(4/2), ±(4/4)
±(2/1), ±(2/2), ±(2/4)
±(1/1), ±(1/2), ±(1/4)
That's a lot of possibilities, so it might help to divide through by 4, leaving
P(x) = 4(x^4 - 1/2 x^3 + 1/4 x^2 - 3)
Now we know that any rational roots will be found among ±1,±3