How do you use the rational root theorem to find the roots of #2x^4 -7x^3 -35x^2 +13x +3#?

1 Answer
George C.
Sep 2, 2015

Use rational root theorem to find possible rational roots and try them to find rational roots #-3# and #1/2#.

Then divide #f(x)# by #(x+3)(2x-1)# and solve to get roots:

#3+-sqrt(10)#

Explanation:

Let #f(x) = 2x^4-7x^3-35x^2+13x+3#

By the rational root theorem, any rational roots of #f(x) = 0# must be expressible as #p/q# where #p, q in ZZ#, #q > 0#, #"hcf"(p, q) = 1#, #p | 3#, #q | 2#

So the only possible rational roots are:

#+-3#, #+-3/2#, #+-1#, #+-1/2#

#f(-3) = 162+189-315-39+3 = 0#

#f(1/2) = 1/8-7/8-35/4+13/2+3#

#= (1-7-70+52+24)/8 = 0#

So #x=-3# and #x=1/2# are roots and #f(x)# is divisible by

#(x+3)(2x-1) = 2x^2+5x-3#

Use synthetic division to find:

#2x^4-7x^3-35x^2+13x+3#

#= (2x^2+5x-3)(x^2-6x-1)#

The remaining quadratic factor #(x^2-6x-1)# has irrational roots given by the quadratic formula as:

#x = (6+-sqrt(36+4))/2 = 3+-sqrt(10)#

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