How do you find the rational roots of #x^4+3x^3+3x^2-3x-4=0#?
Find rational roots
First note that the sum of the coefficients is zero (
#x^4+3x^3+3x^2-3x-4 = (x-1)(x^3+4x^2+7x+4)#
The coefficients of the remaining factor are all positive, so it has no positive zeros. Notice that
#x^3+4x^2+7x+4 = (x+1)(x^2+3x+4)#
The remaining quadratic factor is of the form
#Delta = b^2-4ac = 3^2-(4xx1xx4) = 9-16 = -7#
Since this is negative, there are no more linear factors with Real coefficients and no more Real roots, let alone rational ones.
Alternatively, use the rational root theorem to find that the only possible rational roots are:
then just try each one in turn.