Possible and actual rational roots of $$$f{\left(x \right)} = x^{3} - 31 x - 30$$$
Your Input
Find the rational zeros of $$$x^{3} - 31 x - 30 = 0$$$.
Solution
Since all coefficients are integers, we can apply the rational zeros theorem.
The trailing coefficient (the coefficient of the constant term) is $$$-30$$$.
Find its factors (with the plus sign and the minus sign): $$$\pm 1$$$, $$$\pm 2$$$, $$$\pm 3$$$, $$$\pm 5$$$, $$$\pm 6$$$, $$$\pm 10$$$, $$$\pm 15$$$, $$$\pm 30$$$.
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 $$$\frac{p}{q}$$$: $$$\pm \frac{1}{1}$$$, $$$\pm \frac{2}{1}$$$, $$$\pm \frac{3}{1}$$$, $$$\pm \frac{5}{1}$$$, $$$\pm \frac{6}{1}$$$, $$$\pm \frac{10}{1}$$$, $$$\pm \frac{15}{1}$$$, $$$\pm \frac{30}{1}$$$.
Simplify and remove the duplicates (if any).
These are the possible rational roots: $$$\pm 1$$$, $$$\pm 2$$$, $$$\pm 3$$$, $$$\pm 5$$$, $$$\pm 6$$$, $$$\pm 10$$$, $$$\pm 15$$$, $$$\pm 30$$$.
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 $$$x^{3} - 31 x - 30$$$ by $$$x - 1$$$.
$$$P{\left(1 \right)} = -60$$$; thus, the remainder is $$$-60$$$.
Check $$$-1$$$: divide $$$x^{3} - 31 x - 30$$$ by $$$x - \left(-1\right) = x + 1$$$.
$$$P{\left(-1 \right)} = 0$$$; thus, the remainder is $$$0$$$.
Hence, $$$-1$$$ is a root.
Check $$$2$$$: divide $$$x^{3} - 31 x - 30$$$ by $$$x - 2$$$.
$$$P{\left(2 \right)} = -84$$$; thus, the remainder is $$$-84$$$.
Check $$$-2$$$: divide $$$x^{3} - 31 x - 30$$$ by $$$x - \left(-2\right) = x + 2$$$.
$$$P{\left(-2 \right)} = 24$$$; thus, the remainder is $$$24$$$.
Check $$$3$$$: divide $$$x^{3} - 31 x - 30$$$ by $$$x - 3$$$.
$$$P{\left(3 \right)} = -96$$$; thus, the remainder is $$$-96$$$.
Check $$$-3$$$: divide $$$x^{3} - 31 x - 30$$$ by $$$x - \left(-3\right) = x + 3$$$.
$$$P{\left(-3 \right)} = 36$$$; thus, the remainder is $$$36$$$.
Check $$$5$$$: divide $$$x^{3} - 31 x - 30$$$ by $$$x - 5$$$.
$$$P{\left(5 \right)} = -60$$$; thus, the remainder is $$$-60$$$.
Check $$$-5$$$: divide $$$x^{3} - 31 x - 30$$$ by $$$x - \left(-5\right) = x + 5$$$.
$$$P{\left(-5 \right)} = 0$$$; thus, the remainder is $$$0$$$.
Hence, $$$-5$$$ is a root.
Check $$$6$$$: divide $$$x^{3} - 31 x - 30$$$ by $$$x - 6$$$.
$$$P{\left(6 \right)} = 0$$$; thus, the remainder is $$$0$$$.
Hence, $$$6$$$ is a root.
Check $$$-6$$$: divide $$$x^{3} - 31 x - 30$$$ by $$$x - \left(-6\right) = x + 6$$$.
$$$P{\left(-6 \right)} = -60$$$; thus, the remainder is $$$-60$$$.
Check $$$10$$$: divide $$$x^{3} - 31 x - 30$$$ by $$$x - 10$$$.
$$$P{\left(10 \right)} = 660$$$; thus, the remainder is $$$660$$$.
Check $$$-10$$$: divide $$$x^{3} - 31 x - 30$$$ by $$$x - \left(-10\right) = x + 10$$$.
$$$P{\left(-10 \right)} = -720$$$; thus, the remainder is $$$-720$$$.
Check $$$15$$$: divide $$$x^{3} - 31 x - 30$$$ by $$$x - 15$$$.
$$$P{\left(15 \right)} = 2880$$$; thus, the remainder is $$$2880$$$.
Check $$$-15$$$: divide $$$x^{3} - 31 x - 30$$$ by $$$x - \left(-15\right) = x + 15$$$.
$$$P{\left(-15 \right)} = -2940$$$; thus, the remainder is $$$-2940$$$.
Check $$$30$$$: divide $$$x^{3} - 31 x - 30$$$ by $$$x - 30$$$.
$$$P{\left(30 \right)} = 26040$$$; thus, the remainder is $$$26040$$$.
Check $$$-30$$$: divide $$$x^{3} - 31 x - 30$$$ by $$$x - \left(-30\right) = x + 30$$$.
$$$P{\left(-30 \right)} = -26100$$$; thus, the remainder is $$$-26100$$$.
Answer
Possible rational roots: $$$\pm 1$$$, $$$\pm 2$$$, $$$\pm 3$$$, $$$\pm 5$$$, $$$\pm 6$$$, $$$\pm 10$$$, $$$\pm 15$$$, $$$\pm 30$$$A.
Actual rational roots: $$$-1$$$, $$$-5$$$, $$$6$$$A.