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.
$p(x) = -6x - 6 - 48x^3 + 60x^2$
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: Rearrange polynomial in standard form

$p(x) = -48x^3 + 60x^2 -6x -6$

Step2: Simplify polynomial (divide by -6)

$8x^3 -10x^2 +x +1 =0$

Step3: Apply Rational Root Theorem

Possible roots: $\pm\frac{\text{factors of constant term }1}{\text{factors of leading coefficient }8}$
$\pm1, \pm\frac{1}{2}, \pm\frac{1}{4}, \pm\frac{1}{8}$

Step4: Test possible roots via substitution

Test $x=1$: $8(1)^3 -10(1)^2 +1 +1 = 8-10+1+1=0$, so $x=1$ is a root.

Step5: Factor polynomial using root $x=1$

Use synthetic division to divide $8x^3 -10x^2 +x +1$ by $(x-1)$:
Quotient: $8x^2 -2x -1$

Step6: Solve quadratic for remaining roots

Factor $8x^2 -2x -1$:
$(4x+1)(2x-1)=0$
Solve for $x$: $4x+1=0 \implies x=-\frac{1}{4}$; $2x-1=0 \implies x=\frac{1}{2}$

Answer:

Possible rational roots: $1, -1, \frac{1}{2}, -\frac{1}{2}, \frac{1}{4}, -\frac{1}{4}, \frac{1}{8}, -\frac{1}{8}$
Actual roots: $1, \frac{1}{2}, -\frac{1}{4}$