Possible and actual rational roots of $$$f{\left(x \right)} = x^{4} + 8 x^{3} + 11 x^{2} + 9 x + 28$$$

Find the rational zeros of $$$x^{4} + 8 x^{3} + 11 x^{2} + 9 x + 28 = 0$$$.

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

The trailing coefficient (the coefficient of the constant term) is $$$28$$$.

Find its factors (with the plus sign and the minus sign): $$$\pm 1$$$, $$$\pm 2$$$, $$$\pm 4$$$, $$$\pm 7$$$, $$$\pm 14$$$, $$$\pm 28$$$.

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 $$$\frac{p}{q}$$$: $$$\pm \frac{1}{1}$$$, $$$\pm \frac{2}{1}$$$, $$$\pm \frac{4}{1}$$$, $$$\pm \frac{7}{1}$$$, $$$\pm \frac{14}{1}$$$, $$$\pm \frac{28}{1}$$$.

Simplify and remove the duplicates (if any).

These are the possible rational roots: $$$\pm 1$$$, $$$\pm 2$$$, $$$\pm 4$$$, $$$\pm 7$$$, $$$\pm 14$$$, $$$\pm 28$$$.

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 $$$x^{4} + 8 x^{3} + 11 x^{2} + 9 x + 28$$$ by $$$x - 1$$$.

  $$$P{\left(1 \right)} = 57$$$; thus, the remainder is $$$57$$$.

• Check $$$-1$$$: divide $$$x^{4} + 8 x^{3} + 11 x^{2} + 9 x + 28$$$ by $$$x - \left(-1\right) = x + 1$$$.

  $$$P{\left(-1 \right)} = 23$$$; thus, the remainder is $$$23$$$.

• Check $$$2$$$: divide $$$x^{4} + 8 x^{3} + 11 x^{2} + 9 x + 28$$$ by $$$x - 2$$$.

  $$$P{\left(2 \right)} = 170$$$; thus, the remainder is $$$170$$$.

• Check $$$-2$$$: divide $$$x^{4} + 8 x^{3} + 11 x^{2} + 9 x + 28$$$ by $$$x - \left(-2\right) = x + 2$$$.

  $$$P{\left(-2 \right)} = 6$$$; thus, the remainder is $$$6$$$.

• Check $$$4$$$: divide $$$x^{4} + 8 x^{3} + 11 x^{2} + 9 x + 28$$$ by $$$x - 4$$$.

  $$$P{\left(4 \right)} = 1008$$$; thus, the remainder is $$$1008$$$.

• Check $$$-4$$$: divide $$$x^{4} + 8 x^{3} + 11 x^{2} + 9 x + 28$$$ by $$$x - \left(-4\right) = x + 4$$$.

  $$$P{\left(-4 \right)} = -88$$$; thus, the remainder is $$$-88$$$.

• Check $$$7$$$: divide $$$x^{4} + 8 x^{3} + 11 x^{2} + 9 x + 28$$$ by $$$x - 7$$$.

  $$$P{\left(7 \right)} = 5775$$$; thus, the remainder is $$$5775$$$.

• Check $$$-7$$$: divide $$$x^{4} + 8 x^{3} + 11 x^{2} + 9 x + 28$$$ by $$$x - \left(-7\right) = x + 7$$$.

  $$$P{\left(-7 \right)} = 161$$$; thus, the remainder is $$$161$$$.

• Check $$$14$$$: divide $$$x^{4} + 8 x^{3} + 11 x^{2} + 9 x + 28$$$ by $$$x - 14$$$.

  $$$P{\left(14 \right)} = 62678$$$; thus, the remainder is $$$62678$$$.

• Check $$$-14$$$: divide $$$x^{4} + 8 x^{3} + 11 x^{2} + 9 x + 28$$$ by $$$x - \left(-14\right) = x + 14$$$.

  $$$P{\left(-14 \right)} = 18522$$$; thus, the remainder is $$$18522$$$.

• Check $$$28$$$: divide $$$x^{4} + 8 x^{3} + 11 x^{2} + 9 x + 28$$$ by $$$x - 28$$$.

  $$$P{\left(28 \right)} = 799176$$$; thus, the remainder is $$$799176$$$.

• Check $$$-28$$$: divide $$$x^{4} + 8 x^{3} + 11 x^{2} + 9 x + 28$$$ by $$$x - \left(-28\right) = x + 28$$$.

  $$$P{\left(-28 \right)} = 447440$$$; thus, the remainder is $$$447440$$$.

Possible rational roots: $$$\pm 1$$$, $$$\pm 2$$$, $$$\pm 4$$$, $$$\pm 7$$$, $$$\pm 14$$$, $$$\pm 28$$$A.

Actual rational roots: no rational roots.