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Possible and actual rational roots of $$$f{\left(x \right)} = x^{3} - x^{2} - 17 x - 15$$$
Your Input
Find the rational zeros of $$$x^{3} - x^{2} - 17 x - 15 = 0$$$.
Solution
Since all coefficients are integers, we can apply the rational zeros theorem.
The trailing coefficient (the coefficient of the constant term) is $$$-15$$$.
Find its factors (with the plus sign and the minus sign): $$$\pm 1$$$, $$$\pm 3$$$, $$$\pm 5$$$, $$$\pm 15$$$.
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 $$$\frac{p}{q}$$$: $$$\pm \frac{1}{1}$$$, $$$\pm \frac{3}{1}$$$, $$$\pm \frac{5}{1}$$$, $$$\pm \frac{15}{1}$$$.
Simplify and remove the duplicates (if any).
These are the possible rational roots: $$$\pm 1$$$, $$$\pm 3$$$, $$$\pm 5$$$, $$$\pm 15$$$.
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 $$$x^{3} - x^{2} - 17 x - 15$$$ by $$$x - 1$$$.
$$$P{\left(1 \right)} = -32$$$; thus, the remainder is $$$-32$$$.
Check $$$-1$$$: divide $$$x^{3} - x^{2} - 17 x - 15$$$ by $$$x - \left(-1\right) = x + 1$$$.
$$$P{\left(-1 \right)} = 0$$$; thus, the remainder is $$$0$$$.
Hence, $$$-1$$$ is a root.
Check $$$3$$$: divide $$$x^{3} - x^{2} - 17 x - 15$$$ by $$$x - 3$$$.
$$$P{\left(3 \right)} = -48$$$; thus, the remainder is $$$-48$$$.
Check $$$-3$$$: divide $$$x^{3} - x^{2} - 17 x - 15$$$ by $$$x - \left(-3\right) = x + 3$$$.
$$$P{\left(-3 \right)} = 0$$$; thus, the remainder is $$$0$$$.
Hence, $$$-3$$$ is a root.
Check $$$5$$$: divide $$$x^{3} - x^{2} - 17 x - 15$$$ by $$$x - 5$$$.
$$$P{\left(5 \right)} = 0$$$; thus, the remainder is $$$0$$$.
Hence, $$$5$$$ is a root.
Check $$$-5$$$: divide $$$x^{3} - x^{2} - 17 x - 15$$$ by $$$x - \left(-5\right) = x + 5$$$.
$$$P{\left(-5 \right)} = -80$$$; thus, the remainder is $$$-80$$$.
Check $$$15$$$: divide $$$x^{3} - x^{2} - 17 x - 15$$$ by $$$x - 15$$$.
$$$P{\left(15 \right)} = 2880$$$; thus, the remainder is $$$2880$$$.
Check $$$-15$$$: divide $$$x^{3} - x^{2} - 17 x - 15$$$ by $$$x - \left(-15\right) = x + 15$$$.
$$$P{\left(-15 \right)} = -3360$$$; thus, the remainder is $$$-3360$$$.
Answer
Possible rational roots: $$$\pm 1$$$, $$$\pm 3$$$, $$$\pm 5$$$, $$$\pm 15$$$A.
Actual rational roots: $$$-1$$$, $$$-3$$$, $$$5$$$A.