How do you use the rational roots theorem to find all possible zeros of f(x)=x424x225?

1 Answer
Mar 2, 2016

All possible zeros of f(x)=x424x225 are {5,5,i,i}.

Explanation:

As the function f(x)=x424x225 contains only even powers of x, it can be easily factorized by splitting middle term 24x2 as 25x2+x2. Hence, x424x225 becomes

x425x2+x225

= x2(x225)+1(x225)

= (x2+1)(x225) - note that second term is of form a2b2 and f(x) can be further factorized as

(x2+1)(x+5)(x5)

Also note that discriminant of x2+1 is negative and so if domain is real numbers, the only roots of f(x)=0 are given by x+5=0 and x5=0 i.e. x=5 and x=5.

However, if domain is complex numbers, as x2+1=0, we can also include i and i as roots.

Hence, all possible zeros of f(x)=x424x225 are {5,5,i,i}.