Question 1206504


1. 

{{{x^3 + x^2 + 4x = -4}}}
{{{x^3 + x^2 + 4x +4=0}}}
{{{(x^3 + x^2)+ (4x +4)=0}}}
{{{x^2(x + 1)+ 4(x +1)=0}}}
{{{(x^2+ 4)(x +1)=0}}}

roots:
if {{{x^2+ 4=0 }}}=>{{{x^2=-4}}}=>{{{x=sqrt(-4)}}}=>{{{x=2i}}} or{{{ x=-2i}}}

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

solutions:
{{{x=2i}}}
{{{x=-2i}}}
{{{x=-1}}}



2. 

{{{2x^3 - 9x^2 - 11x + 8 = 0}}}

the coefficient of the constant term is {{{8}}}

find its factors (with the plus sign and the minus sign): ±{{{1}}}, ±{{{2}}}, ±{{{4}}}, ±{{{8}}}
these are the possible values for{{{ p}}}

the leading coefficient is {{{2}}}
 its factors : ±{{{1}}}, ±{{{2}}}
these are the possible values for {{{q}}}


find all possible values for{{{ p/q}}}
±{{{1/1}}}, ±{{{1/2}}},±{{{2/1}}},±{{{2/2}}},±{{{4/1}}},±{{{4/2}}}, ±{{{8/1}}},±{{{8/2}}}

±{{{1}}}, ±{{{1/2}}},±{{{2}}},±{{{1}}},±{{{4}}},±{{{2}}}, ±{{{8}}},±{{{4}}}
±{{{1}}}, ±{{{1/2}}},±{{{2}}},±{{{4}}}, ±{{{8}}}


check the possible roots: 

if {{{a}}} is a root of the polynomial {{{P(x)}}}, the remainder from the division of {{{P(x)}}} by x-a should equal {{{0}}}

Check {{{1}}}:
divide {{{(2x^3 - 9x^2 - 11x + 8 )/(x-1)}}}
when you do long division, you will get reminder {{{-10}}}

Check{{{ -1}}}:
divide {{{(2x^3 - 9x^2 - 11x + 8 )/(x+1)}}}
when you do long division, you will get reminder {{{8}}}

Check {{{1/2}}}: the remainder is {{{1/2}}}
​
 Check {{{-1/2}}}: the remainder is {{{11}}}

Check {{{2}}}: the remainder is {{{-34}}}
Check {{{-2}}}: the remainder is{{{-22}}}
 
Check{{{ 4}}}: the remainder is {{{-52}}}
Check {{{-4}}}: the remainder is{{{ -220}}}

Check {{{8}}}: the remainder is {{{368}}}
Check{{{ -8}}}: the remainder is {{{-1504}}}


answer:

possible rational roots: ±{{{1}}}, ±{{{1/2}}},±{{{2}}},±{{{4}}}, ±{{{8}}}
actual rational roots: {{{no}}} rational roots