Possible and actual rational roots of $$$f{\left(x \right)} = x^{6} - 64$$$
Your Input
Find the rational zeros of $$$x^{6} - 64 = 0$$$.
Solution
Since all coefficients are integers, we can apply the rational zeros theorem.
The trailing coefficient (the coefficient of the constant term) is $$$-64$$$.
Find its factors (with the plus sign and the minus sign): $$$\pm 1$$$, $$$\pm 2$$$, $$$\pm 4$$$, $$$\pm 8$$$, $$$\pm 16$$$, $$$\pm 32$$$, $$$\pm 64$$$.
These are the possible values for $$$p$$$.
The leading coefficient (the coefficient of the term with the highest degree) is $$$1$$$.
Find its factors (with the plus sign and the minus sign): $$$\pm 1$$$.
These are the possible values for $$$q$$$.
Find all possible values of $$$\frac{p}{q}$$$: $$$\pm \frac{1}{1}$$$, $$$\pm \frac{2}{1}$$$, $$$\pm \frac{4}{1}$$$, $$$\pm \frac{8}{1}$$$, $$$\pm \frac{16}{1}$$$, $$$\pm \frac{32}{1}$$$, $$$\pm \frac{64}{1}$$$.
Simplify and remove the duplicates (if any).
These are the possible rational roots: $$$\pm 1$$$, $$$\pm 2$$$, $$$\pm 4$$$, $$$\pm 8$$$, $$$\pm 16$$$, $$$\pm 32$$$, $$$\pm 64$$$.
Next, check the possible roots: if $$$a$$$ is a root of the polynomial $$$P{\left(x \right)}$$$, the remainder from the division of $$$P{\left(x \right)}$$$ by $$$x - a$$$ should equal $$$0$$$ (according to the remainder theorem, this means that $$$P{\left(a \right)} = 0$$$).
Check $$$1$$$: divide $$$x^{6} - 64$$$ by $$$x - 1$$$.
$$$P{\left(1 \right)} = -63$$$; thus, the remainder is $$$-63$$$.
Check $$$-1$$$: divide $$$x^{6} - 64$$$ by $$$x - \left(-1\right) = x + 1$$$.
$$$P{\left(-1 \right)} = -63$$$; thus, the remainder is $$$-63$$$.
Check $$$2$$$: divide $$$x^{6} - 64$$$ by $$$x - 2$$$.
$$$P{\left(2 \right)} = 0$$$; thus, the remainder is $$$0$$$.
Hence, $$$2$$$ is a root.
Check $$$-2$$$: divide $$$x^{6} - 64$$$ by $$$x - \left(-2\right) = x + 2$$$.
$$$P{\left(-2 \right)} = 0$$$; thus, the remainder is $$$0$$$.
Hence, $$$-2$$$ is a root.
Check $$$4$$$: divide $$$x^{6} - 64$$$ by $$$x - 4$$$.
$$$P{\left(4 \right)} = 4032$$$; thus, the remainder is $$$4032$$$.
Check $$$-4$$$: divide $$$x^{6} - 64$$$ by $$$x - \left(-4\right) = x + 4$$$.
$$$P{\left(-4 \right)} = 4032$$$; thus, the remainder is $$$4032$$$.
Check $$$8$$$: divide $$$x^{6} - 64$$$ by $$$x - 8$$$.
$$$P{\left(8 \right)} = 262080$$$; thus, the remainder is $$$262080$$$.
Check $$$-8$$$: divide $$$x^{6} - 64$$$ by $$$x - \left(-8\right) = x + 8$$$.
$$$P{\left(-8 \right)} = 262080$$$; thus, the remainder is $$$262080$$$.
Check $$$16$$$: divide $$$x^{6} - 64$$$ by $$$x - 16$$$.
$$$P{\left(16 \right)} = 16777152$$$; thus, the remainder is $$$16777152$$$.
Check $$$-16$$$: divide $$$x^{6} - 64$$$ by $$$x - \left(-16\right) = x + 16$$$.
$$$P{\left(-16 \right)} = 16777152$$$; thus, the remainder is $$$16777152$$$.
Check $$$32$$$: divide $$$x^{6} - 64$$$ by $$$x - 32$$$.
$$$P{\left(32 \right)} = 1073741760$$$; thus, the remainder is $$$1073741760$$$.
Check $$$-32$$$: divide $$$x^{6} - 64$$$ by $$$x - \left(-32\right) = x + 32$$$.
$$$P{\left(-32 \right)} = 1073741760$$$; thus, the remainder is $$$1073741760$$$.
Check $$$64$$$: divide $$$x^{6} - 64$$$ by $$$x - 64$$$.
$$$P{\left(64 \right)} = 68719476672$$$; thus, the remainder is $$$68719476672$$$.
Check $$$-64$$$: divide $$$x^{6} - 64$$$ by $$$x - \left(-64\right) = x + 64$$$.
$$$P{\left(-64 \right)} = 68719476672$$$; thus, the remainder is $$$68719476672$$$.
Answer
Possible rational roots: $$$\pm 1$$$, $$$\pm 2$$$, $$$\pm 4$$$, $$$\pm 8$$$, $$$\pm 16$$$, $$$\pm 32$$$, $$$\pm 64$$$A.
Actual rational roots: $$$2$$$, $$$-2$$$A.