Question

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

Answer 1

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}`.