How do you use the rational root theorem to find the roots of #x^4 + 3x^3 - x^2 - 9x - 6 = 0#?

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Answer 1 · 2015-12-10T17:25:41

The roots are #-1#, #-2#, #sqrt3# and #-sqrt3#

By the rational roots theorem, the possible rational roots of #x^4 + 3x^3 - x^2 - 9x - 6 = 0# are #+-1#, #+-2#, #+-3#, or #+-6#.

By trial (or division) #-1# is a root, so (by the factor theorem) #x-(-1) = x+1# is a factor.

By division or trial and error,

#x^4 + 3x^3 - x^2 - 9x - 6 = (x+1)(x^3+2x^2-3x-6)#

Possible rational roots of #x^3+2x^2-3x-6 = 0# are the same as for the original.

By trial, #1, -1, 2# are not roots, but #-2# is a root. So #x+2# is a factor. Division gets us

#x^3+2x^2-3x-6 = (x+2)(x^2-3)#

The roots of #x^2-3=0# are #+-sqrt3#.