How do you use the rational roots theorem to find all possible zeros of #f(x) = x^4 -24x^2- 25#?

1 Answer
Shwetank Mauria
Mar 2, 2016

All possible zeros of #f(x)=x^4−24x^2−25# are #{5, -5,i,-i}#.

Explanation:

As the function #f(x)=x^4−24x^2−25# contains only even powers of #x#, it can be easily factorized by splitting middle term #-24x^2# as #-25x^2+x^2#. Hence, #x^4−24x^2−25# becomes

#x^4−25x^2+x^2−25#

= #x^2(x^2-25)+1(x^2-25)#

= #(x^2+1)(x^2-25)# - note that second term is of form #a^2-b^2# and #f(x)# can be further factorized as

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

Also note that discriminant of #x^2+1# is negative and so if domain is real numbers, the only roots of #f(x)=0# are given by #x+5=0# and #x-5=0# i.e. #x=-5# and #x=5#.

However, if domain is complex numbers, as #x^2+1=0#, we can also include #i# and #-i# as roots.

Hence, all possible zeros of #f(x)=x^4−24x^2−25# are #{5, -5,i,-i}#.

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