Question

use the rational root theorem to list all possible rational roots of the given polynomial. then use various methods to determine the actual roots of the given polynomial.
$p(x)=x^4 + 4x^3 + 13x^2 + 18x - 136$
enter the possible rational roots of $p$ using a comma-separated list.
using synthetic division or other various methods, determine the actual roots of $p$.

Explanation:

Step1: Identify constant & leading coeff

Constant term: $-136$, Leading coefficient: $1$

Step2: List factors of both values

Factors of $136$: $\pm1, \pm2, \pm4, \pm8, \pm17, \pm34, \pm68, \pm136$
Factors of $1$: $\pm1$

Step3: Apply Rational Root Theorem

Possible rational roots: $\frac{\text{Factors of }-136}{\text{Factors of }1} = \pm1, \pm2, \pm4, \pm8, \pm17, \pm34, \pm68, \pm136$

Step4: Test $x=2$ via synthetic division

$$\begin{array}{r|rrrrr} 2 & 1 & 4 & 13 & 18 & -136 \\ & & 2 & 12 & 50 & 136 \\ \hline & 1 & 6 & 25 & 68 & 0 \end{array}$$

Remainder = 0, so $x=2$ is a root.

Step5: Factor polynomial to cubic

$p(x)=(x-2)(x^3+6x^2+25x+68)$

Step6: Test $x=-4$ on cubic

$$\begin{array}{r|rrrr} -4 & 1 & 6 & 25 & 68 \\ & & -4 & -8 & -68 \\ \hline & 1 & 2 & 17 & 0 \end{array}$$

Remainder = 0, so $x=-4$ is a root.

Step7: Factor to quadratic

$p(x)=(x-2)(x+4)(x^2+2x+17)$

Step8: Solve quadratic with quadratic formula

For $x^2+2x+17=0$, $a=1, b=2, c=17$
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}=\frac{-2\pm\sqrt{4-68}}{2}=\frac{-2\pm\sqrt{-64}}{2}=-1\pm4i$

Answer:

Possible rational roots: $-136, -68, -34, -17, -8, -4, -2, -1, 1, 2, 4, 8, 17, 34, 68, 136$
Actual roots: $2, -4, -1+4i, -1-4i$