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

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

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Answer 1 · 2016-07-13T00:42:30

(see below)

Explanation:

Given
$XXX 3 x^{3} + 9 x^{2} + 4 x + 12$ $color(white)("XXX")3x^3+9x^2+4x+12$

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:
$XXX = 3 x^{2} (x + 3) + 4 (x + 3)$ $color(white)("XXX")=3x^2(x+3) +4(x+3)$

$XXX = (3 x^{2} + 4) (x + 3)$ $color(white)("XXX")=(3x^2+4)(x+3)$

Since $(3 x^{2} + 4) > 0$ $(3x^2+4)>0$ for all Real values of $x$ $x$
1. there are no Real factors of $(3 x^{2} + 4)$ $(3x^2+4)$
$XXX \to$ $color(white)("XXX")rarr$ there are no Real zeros corresponding to the $(3 x^{2} + 4)$ $(3x^2+4)$ term.

The only Real zero comes from $x + 3 = 0 \to x = - 3$ $x+3=0 rarr x=-3$

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

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

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