Possible and actual rational roots of $$$f{\left(x \right)} = 3 x^{4} + x^{3} - 13 x^{2} - 2 x + 9$$$

Since all coefficients are integers, we can apply the rational zeros theorem.

The trailing coefficient (the coefficient of the constant term) is $$$9$$$.

Find its factors (with the plus sign and the minus sign): $$$\pm 1$$$, $$$\pm 3$$$, $$$\pm 9$$$.

These are the possible values for $$$p$$$.

The leading coefficient (the coefficient of the term with the highest degree) is $$$3$$$.

Find its factors (with the plus sign and the minus sign): $$$\pm 1$$$, $$$\pm 3$$$.

These are the possible values for $$$q$$$.

Find all possible values of $$$\frac{p}{q}$$$: $$$\pm \frac{1}{1}$$$, $$$\pm \frac{1}{3}$$$, $$$\pm \frac{3}{1}$$$, $$$\pm \frac{3}{3}$$$, $$$\pm \frac{9}{1}$$$, $$$\pm \frac{9}{3}$$$.

Simplify and remove the duplicates (if any).

These are the possible rational roots: $$$\pm 1$$$, $$$\pm \frac{1}{3}$$$, $$$\pm 3$$$, $$$\pm 9$$$.

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 $$$3 x^{4} + x^{3} - 13 x^{2} - 2 x + 9$$$ by $$$x - 1$$$.

  $$$P{\left(1 \right)} = -2$$$; thus, the remainder is $$$-2$$$.

• Check $$$-1$$$: divide $$$3 x^{4} + x^{3} - 13 x^{2} - 2 x + 9$$$ by $$$x - \left(-1\right) = x + 1$$$.

  $$$P{\left(-1 \right)} = 0$$$; thus, the remainder is $$$0$$$.

  Hence, $$$-1$$$ is a root.

• Check $$$\frac{1}{3}$$$: divide $$$3 x^{4} + x^{3} - 13 x^{2} - 2 x + 9$$$ by $$$x - \frac{1}{3}$$$.

  $$$P{\left(\frac{1}{3} \right)} = \frac{188}{27}$$$; thus, the remainder is $$$\frac{188}{27}$$$.

• Check $$$- \frac{1}{3}$$$: divide $$$3 x^{4} + x^{3} - 13 x^{2} - 2 x + 9$$$ by $$$x - \left(- \frac{1}{3}\right) = x + \frac{1}{3}$$$.

  $$$P{\left(- \frac{1}{3} \right)} = \frac{74}{9}$$$; thus, the remainder is $$$\frac{74}{9}$$$.

• Check $$$3$$$: divide $$$3 x^{4} + x^{3} - 13 x^{2} - 2 x + 9$$$ by $$$x - 3$$$.

  $$$P{\left(3 \right)} = 156$$$; thus, the remainder is $$$156$$$.

• Check $$$-3$$$: divide $$$3 x^{4} + x^{3} - 13 x^{2} - 2 x + 9$$$ by $$$x - \left(-3\right) = x + 3$$$.

  $$$P{\left(-3 \right)} = 114$$$; thus, the remainder is $$$114$$$.

• Check $$$9$$$: divide $$$3 x^{4} + x^{3} - 13 x^{2} - 2 x + 9$$$ by $$$x - 9$$$.

  $$$P{\left(9 \right)} = 19350$$$; thus, the remainder is $$$19350$$$.

• Check $$$-9$$$: divide $$$3 x^{4} + x^{3} - 13 x^{2} - 2 x + 9$$$ by $$$x - \left(-9\right) = x + 9$$$.

  $$$P{\left(-9 \right)} = 17928$$$; thus, the remainder is $$$17928$$$.

Possible rational roots: $$$\pm 1$$$, $$$\pm \frac{1}{3}$$$, $$$\pm 3$$$, $$$\pm 9$$$A.

Actual rational root: $$$-1$$$A.