How do you use the rational roots theorem to find all possible zeros of #f(x)=x^4-x-4#?

1 Answer
Mar 23, 2016

Answer:

The rational root theorem helps us determine that this #f(x)# has no rational zeros, only irrational and/or Complex ones.

Explanation:

#f(x) = x^4-x-4#

By the rational roots theorem, any rational zeros of #f(x)# must be expressible in the form #p/q# for integers #p#, #q# with #p# a divisor of the constant term #-4# and #q# a divisor of the coefficient #1# of the leading term.

That means that the only possible rational zeros are:

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

Trying each in turn, we find:

#f(1) = 1-1-4 = -4#

#f(-1) = 1+1-4 = -2#

#f(2) = 16-2-4 = 10#

#f(-2) = 16+2-4 = 14#

#f(4) = 256-4-4 = 248#

#f(-4) = 256+4-4 = 256#

So there are no rational zeros, but #f(x)# changes sign in #(-2, -1)# and #(1, 2)#, so there are irrational zeros in those intervals.

That's as much as we can learn about this #f(x)# from the rational root theorem.