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

Mar 2, 2016

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

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 $- 24 {x}^{2}$ as $- 25 {x}^{2} + {x}^{2}$. Hence, x^4−24x^2−25 becomes

x^4−25x^2+x^2−25

= ${x}^{2} \left({x}^{2} - 25\right) + 1 \left({x}^{2} - 25\right)$

= $\left({x}^{2} + 1\right) \left({x}^{2} - 25\right)$ - note that second term is of form ${a}^{2} - {b}^{2}$ and $f \left(x\right)$ can be further factorized as

$\left({x}^{2} + 1\right) \left(x + 5\right) \left(x - 5\right)$

Also note that discriminant of ${x}^{2} + 1$ is negative and so if domain is real numbers, the only roots of $f \left(x\right) = 0$ 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 $\left\{5 , - 5 , i , - i\right\}$.