Question 997336
{{{f(x)=x^3+7x^2+4x-12}}}

if {{{x[1]=-2}}} is a zero, means {{{f(x)=x^3+7x^2+4x-12}}} is divisible by {{{x-x[1]=x-(-2)=x+2}}}->an explanation to why {{{x-2}}} is not a factor

use long division:

---------({{{x^2+5x-6}}}
{{{x+2}}}|{{{x^3+7x^2+4x-12}}}
----------{{{x^3+2x^2}}}.......subtract
------------{{{0+5x^2}}}......add {{{4x}}}
------------- {{{5x^2+4x}}}......
------------- {{{5x^2+10x}}}.......subtract
-------------- {{{0-6x}}}.......add {{{-12}}}
--------------- {{{-6x-12}}}.......
--------------- {{{-6x-12}}}.......subtract
----------------- {{{0}}}.......reminder

so, now we know two factors of given polynomial, and they are {{{x+2}}} and {{{x^2+5x-6}}}

so, you have
{{{f(x)=(x+2)(x^2+5x-6)}}} ...now we can factor {{{x^2+5x-6}}} too

{{{f(x)=(x+2)(x^2+6x-x-6)}}}

{{{f(x)=(x+2)((x^2+6x)-(x+6))}}}

{{{f(x)=(x+2)(x(x+6)-(x+6))}}}

{{{f(x)=(x+2)(x-1)(x+6)}}}


so, zeros are:

if {{{0=(x+2)}}}=>{{{x=-2}}}
if {{{0=(x-1)}}}=>{{{x=1}}}
if {{{0=(x+6)}}}=>{{{x=-6}}}


{{{ graph( 600, 600, -10, 10, -15, 15, x^3+7x^2+4x-12) }}}