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Possible and actual rational roots of $$$f{\left(x \right)} = 2 x^{3} - 15 x^{2} + 9 x + 22$$$
Your Input
Find the rational zeros of $$$2 x^{3} - 15 x^{2} + 9 x + 22 = 0$$$.
Solution
Since all coefficients are integers, we can apply the rational zeros theorem.
The trailing coefficient (the coefficient of the constant term) is $$$22$$$.
Find its factors (with the plus sign and the minus sign): $$$\pm 1$$$, $$$\pm 2$$$, $$$\pm 11$$$, $$$\pm 22$$$.
These are the possible values for $$$p$$$.
The leading coefficient (the coefficient of the term with the highest degree) is $$$2$$$.
Find its factors (with the plus sign and the minus sign): $$$\pm 1$$$, $$$\pm 2$$$.
These are the possible values for $$$q$$$.
Find all possible values of $$$\frac{p}{q}$$$: $$$\pm \frac{1}{1}$$$, $$$\pm \frac{1}{2}$$$, $$$\pm \frac{2}{1}$$$, $$$\pm \frac{2}{2}$$$, $$$\pm \frac{11}{1}$$$, $$$\pm \frac{11}{2}$$$, $$$\pm \frac{22}{1}$$$, $$$\pm \frac{22}{2}$$$.
Simplify and remove the duplicates (if any).
These are the possible rational roots: $$$\pm 1$$$, $$$\pm \frac{1}{2}$$$, $$$\pm 2$$$, $$$\pm 11$$$, $$$\pm \frac{11}{2}$$$, $$$\pm 22$$$.
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 $$$2 x^{3} - 15 x^{2} + 9 x + 22$$$ by $$$x - 1$$$.
$$$P{\left(1 \right)} = 18$$$; thus, the remainder is $$$18$$$.
Check $$$-1$$$: divide $$$2 x^{3} - 15 x^{2} + 9 x + 22$$$ by $$$x - \left(-1\right) = x + 1$$$.
$$$P{\left(-1 \right)} = -4$$$; thus, the remainder is $$$-4$$$.
Check $$$\frac{1}{2}$$$: divide $$$2 x^{3} - 15 x^{2} + 9 x + 22$$$ by $$$x - \frac{1}{2}$$$.
$$$P{\left(\frac{1}{2} \right)} = 23$$$; thus, the remainder is $$$23$$$.
Check $$$- \frac{1}{2}$$$: divide $$$2 x^{3} - 15 x^{2} + 9 x + 22$$$ by $$$x - \left(- \frac{1}{2}\right) = x + \frac{1}{2}$$$.
$$$P{\left(- \frac{1}{2} \right)} = \frac{27}{2}$$$; thus, the remainder is $$$\frac{27}{2}$$$.
Check $$$2$$$: divide $$$2 x^{3} - 15 x^{2} + 9 x + 22$$$ by $$$x - 2$$$.
$$$P{\left(2 \right)} = -4$$$; thus, the remainder is $$$-4$$$.
Check $$$-2$$$: divide $$$2 x^{3} - 15 x^{2} + 9 x + 22$$$ by $$$x - \left(-2\right) = x + 2$$$.
$$$P{\left(-2 \right)} = -72$$$; thus, the remainder is $$$-72$$$.
Check $$$11$$$: divide $$$2 x^{3} - 15 x^{2} + 9 x + 22$$$ by $$$x - 11$$$.
$$$P{\left(11 \right)} = 968$$$; thus, the remainder is $$$968$$$.
Check $$$-11$$$: divide $$$2 x^{3} - 15 x^{2} + 9 x + 22$$$ by $$$x - \left(-11\right) = x + 11$$$.
$$$P{\left(-11 \right)} = -4554$$$; thus, the remainder is $$$-4554$$$.
Check $$$\frac{11}{2}$$$: divide $$$2 x^{3} - 15 x^{2} + 9 x + 22$$$ by $$$x - \frac{11}{2}$$$.
$$$P{\left(\frac{11}{2} \right)} = - \frac{99}{2}$$$; thus, the remainder is $$$- \frac{99}{2}$$$.
Check $$$- \frac{11}{2}$$$: divide $$$2 x^{3} - 15 x^{2} + 9 x + 22$$$ by $$$x - \left(- \frac{11}{2}\right) = x + \frac{11}{2}$$$.
$$$P{\left(- \frac{11}{2} \right)} = -814$$$; thus, the remainder is $$$-814$$$.
Check $$$22$$$: divide $$$2 x^{3} - 15 x^{2} + 9 x + 22$$$ by $$$x - 22$$$.
$$$P{\left(22 \right)} = 14256$$$; thus, the remainder is $$$14256$$$.
Check $$$-22$$$: divide $$$2 x^{3} - 15 x^{2} + 9 x + 22$$$ by $$$x - \left(-22\right) = x + 22$$$.
$$$P{\left(-22 \right)} = -28732$$$; thus, the remainder is $$$-28732$$$.
Answer
Possible rational roots: $$$\pm 1$$$, $$$\pm \frac{1}{2}$$$, $$$\pm 2$$$, $$$\pm 11$$$, $$$\pm \frac{11}{2}$$$, $$$\pm 22$$$A.
Actual rational roots: no rational roots.