Question

if $f(x) = x^3 + 9x^2 + 20x + 12$ and $f(-6) = 0$, then find all of the zeros of $f(x)$ algebraically.

Explanation:

Step1: Set up synthetic division

Use root $x=-6$ for $f(x)=x^3+9x^2+20x+12$:

$$\begin{array}{r|rrrr} -6 & 1 & 9 & 20 & 12 \\ & & -6 & -18 & -12 \\ \hline & 1 & 3 & 2 & 0 \end{array}$$

Step2: Get quadratic factor

From the result, the quadratic is $x^2+3x+2$.

Step3: Factor quadratic

Factor the quadratic:
$x^2+3x+2=(x+2)(x+1)$

Step4: Find all zeros

Set factors equal to 0:
$x=-6$, $x=-2$, $x=-1$

Answer:

$x = -6, -2, -1$