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

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Answer 1 · 2015-09-22T15:36:01

Use the rational zero theorem to find one root. Then factor (or divide) and find the other roots.

Possible rational roots: #+-1, +-2, +-1/2, +-1/4#

#1# is a root, so #x-1# is a factor on the polynomial.
By division or by stepwise analysis,

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

# = 4x^2(x-1)+x(x-1)+2x-2#

# = 4x^2(x-1)+x(x-1)+2(x-1)#

# = (x-1)(4x^2+x+2)#

The other two roots are roots of #4x^2+x+2 = 0#, which may be found by completing the square or by the quadratic formula.

The roots of the original equation are:

#1, (-1+sqrt31 i)/8, (-1-sqrt31 i)/8#