Question 1090305
P(x)=x^3+4x^2+x-6 


1. Arrange the polynomial in descending order: 

{{{P(x)=x^3+4x^2+x-6 }}}

2. Write down all the factors of the constant term. These are all the possible values of p . 

{{{p=6}}}, all the factors  are

{{{p}}}=  {{{1}}},± {{{2}}},{{{3}}},{{{6}}}

3. Write down all the factors of the leading coefficient. These are all the possible values of q .

{{{q=1}}} 


4.Write down all the possible values of {{{p/q}}}. Remember that since factors can be negative, {{{p/q }}}and {{{- p/q}}} must both be included. Simplify each value and cross out any duplicates. 

 {{{p/q}}}=±{{{1/1}}}, ±{{{2/1}}},±{{{3/1}}}, ±{{{6/1}}},... (eliminate {{{6/1}}}, it is product of {{{2/1}}} and {{{3/1}}})

{{{p/q}}}=±{{{1}}}, ±{{{2}}},±{{{3}}}, 

5. Use synthetic division to determine the values of {{{ p/q}}} for which {{{P(p/q) = 0}}} . These are all the rational roots of {{{P(x) }}}. 

The synthetic division table is: if {{{p/q= 1}}}
{{{1}}}|....  {{{1}}}....   {{{ 4}}}.... ..{{{1}}}.... {{{ -6}}}
...................{{{1}}}.... ..{{{5}}}.... ..{{{6}}}
-----------------------------------------
.........{{{1}}}........{{{5}}}.......{{{6}}}........{{{0}}}-> reminder=> {{{0}}},=>  {{{highlight(1)}}} {{{is}}} a root


The synthetic division table is: if {{{p/q= -1}}}
{{{-1}}}|....  {{{1}}}....    {{{4}}}.... ..{{{1}}}.... {{{ -6}}}
...................{{{-1}}}.... {{{-3}}}.... ....{{{ 2}}}
-----------------------------------------
.........{{{1}}}........{{{3}}}......{{{-2}}}....{{{-4}}}-> reminder {{{-4}}}=> {{{-1}}} is {{{not}}} root

The synthetic division table is: if {{{p/q= 2}}}
{{{2}}}|....  {{{1}}}....    {{{4}}}.... ..{{{1}}}.... {{{ -6}}}
...................{{{2}}}.... ..{{{12}}}....{{{ 26}}}
-----------------------------------------
.........{{{1}}}........{{{6}}}.......{{{13}}}........{{{20}}}-> reminder=> {{{20}}},=>  {{{2}}} is {{{not}}} a root


The synthetic division table is: if {{{p/q= -2}}}
 {{{-2}}}|....   {{{1}}}....    {{{ 4}}}.... .. {{{1}}}....  {{{ -6}}}
................... {{{-2}}}.... .. {{{-4}}}.... {{{6}}}
-----------------------------------------
......... {{{1}}}........ {{{2}}}....... {{{-3}}}....... {{{0}}}-> reminder=> {{{0}}},=>  {{{highlight(-2)}}} {{{is}}} a root


The synthetic division table is: if {{{p/q= 3}}}
 {{{3}}}|....   {{{1}}}....    {{{ 4}}}.... .. {{{1}}}....  {{{ -6}}}
................... {{{3}}}.... .. {{{21}}}....  {{{66}}}
-----------------------------------------
......... {{{1}}}........ {{{7}}}....... {{{22}}}........ {{{60}}}-> reminder=> {{{60}}},=>  {{{3}}} is {{{not}}} a root


The synthetic division table is: if {{{p/q= -3}}}
{{{-3}}}|....  {{{1}}}....    {{{4}}}.... ..{{{1}}}.... {{{ -6}}}
...................{{{-3}}}.... ..{{{-3}}}.... {{{6}}}
-----------------------------------------
.........{{{1}}}........{{{1}}}.......{{{-2}}}........{{{0}}}-> reminder=> {{{0}}},=>  {{{highlight(-3)}}}  {{{is}}} a root

 {{{P(x)=(x - 1) (x + 2) (x + 3)}}}

then solve{{{ P(x)=0 }}}

{{{(x - 1) (x + 2) (x + 3)=0}}}

if {{{x - 1=0}}}=>{{{x=1}}}
if {{{x + 2=0}}}=>{{{x=-2}}}
if {{{x + 3=0}}}=>{{{x=-3}}}