# How do you list all possible roots and find all factors and zeroes of 3x^3+9x^2+4x+12?

Jul 13, 2016

(see below)

#### Explanation:

Given
$\textcolor{w h i t e}{\text{XXX}} 3 {x}^{3} + 9 {x}^{2} + 4 x + 12$

Noting that the ratio of the constants for the first two terms is the same as the ratio for the last two terms, provides a hint:
$\textcolor{w h i t e}{\text{XXX}} = 3 {x}^{2} \left(x + 3\right) + 4 \left(x + 3\right)$

$\textcolor{w h i t e}{\text{XXX}} = \left(3 {x}^{2} + 4\right) \left(x + 3\right)$

Since $\left(3 {x}^{2} + 4\right) > 0$ for all Real values of $x$
1. there are no Real factors of $\left(3 {x}^{2} + 4\right)$
$\textcolor{w h i t e}{\text{XXX}} \rightarrow$ there are no Real zeros corresponding to the $\left(3 {x}^{2} + 4\right)$ term.

1. The only Real zero comes from $x + 3 = 0 \rightarrow x = - 3$

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If Complex values are allowed:
$\left(3 {x}^{2} + 4\right) = 3 \left(x - \frac{2}{\sqrt{3}} i\right) \left(x + \frac{2}{\sqrt{3}} i\right)$ as further factoring
and
complex zeroes at $x = \frac{2}{\sqrt{3}} i$ and $x = - \frac{2}{\sqrt{3}} i$