Question

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

Answer 1

(see below)

Explanation:

Given
`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:
`color(white)("XXX")=3x^2(x+3) +4(x+3)`

`color(white)("XXX")=(3x^2+4)(x+3)`

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

1. The only Real zero comes from `x+3=0 rarr x=-3`

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