How do you use the rational roots theorem to find all possible zeros of # P(x) = 2x^5 + 3x^4 + 2x^2 - 2#?

1 Answer
Hubert
Apr 25, 2016

We seek for rational roots within numbers #p/q#, where #p | a_0# and #q | a_n#.
If #p | -2# then #p in {+-1,+-2}#.
If #q | 2# then #q in {+-1,+-2}#.
Thus #p/q in R={+-1,+-1/2,+-2}#.

Now, if the polynomian #P(x)# has any rational roots, they all are in #R# so we proceed by trial and error:

#P(1)=5 !=0#
#P(-1)=1 !=0#
#P(0.5)=-1.25 !=0#
#P(-0.5)=-1.375 !=0#
#P(2)=118 !=0#
#P(-2)=-10 !=0#

From this we can conclude that the polynomial #P(x)# has no rational roots.

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