How do you use the rational root theorem to find the roots of #x^4-x^3-2x^2-4x-24#?

1 Answer
Sep 6, 2015

Answer:

Use rational root theorem to find possible rational zeros. Try the first few, then use synthetic division with the roots found to find there are no more Real ones.

#x = -2# or #x = 3#

Explanation:

Let #f(x) = x^4-x^3-2x^2-4x-24#

By the rational root theorem, any rational roots of #f(x) = 0# must be expressible in the form #p/q# in lowest terms, where #p, q in ZZ#, #q > 0#, #p# a divisor of the constant term #-24# and #q# a divisor of the coefficient #1# of the leading term.

So the only possible rational roots are:

#+-1#, #+-2#, #+-3#, #+-4#, #+-6#, #+-8#, #+-12# and #+-24#.

Let us try a few:

#f(1) = 1-1-2-4-24 = -30#
#f(-1) = 1+1-2+4-24 = -20#
#f(2) = 16-8-8-8-24 = -32#
#color(blue)(f(-2) = 16+8-8+8-24 = 0)#
#color(blue)(f(3) = 81-27-18-12-24 = 0)#

So #-2# and #3# are roots. Before going further, let's divide #f(x)# by the corresponding factors #(x+2)# and #(x-3)#

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

Use synthetic division:
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So #x^4-x^3-2x^2-4x-24 = (x^2-x-6)(x^2+4)#

#x^2+4# has no Real zeros since #x^2 >= 0# for all #x in RR#