How do you use the rational root theorem to find the roots of # f(x) = x^3-4x^2-2x+8#?

1 Answer
Dec 6, 2015

Answer:

Use the rational root theorem to identify the possible rational roots to try. Hence find #x = 4# and hence the other two irrational roots #x = +-sqrt(2)#

Explanation:

#f(x) = x^3-4x^2-2x+8#

By the rational root theorem, any rational roots of #f(x) = 0# must be expressible in lowest terms in the form #p/q# for some integers #p#, #q# where #p# is a divisor of the constant term #8# and #q# is a divisor of the coefficient #1# of the leading term.

So the only possible rational roots are:

#+-1#, #+-2#, #+-4#, #+-8#

#+-1# won't work since the leading term would be odd and the remaining terms are even.

#f(2) = 8-16-4+8 = -4#

#f(-2) = -8-16+4+8 = -12#

#f(4) = 64 - 64 - 8 + 8 = 0#

So #x = 4# is a root and #(x-4)# a factor.

#x^3-4x^2-2x+8 = (x-4)(x^2-2)#

The remaining roots are the irrational roots of #x^2-2 = 0#, that is:

#x = +-sqrt(2)#