Question 1187763

use long division

......({{{x^2+(a +1)x+(b+a +1)}}}
 {{{x-1}}}| {{{x^3+ax^2+bx-3}}}
........{{{x^3-x^2}}}
..............{{{ax^2+x^2}}}
..............{{{(a +1)x^2+bx}}}
..............{{{(a +1)x^2-(a +1)x }}}
..............................{{{bx+(a +1)x -3}}}
................................{{{(b+a +1)x -(b+a +1)}}}
........................................... {{{-3+b+a +1}}}
........................................... {{{ b+a -2 }}}=> reminder{{{ b+a -2 =1}}}=>{{{b+a =3}}}..........eq.1



......({{{x^2+(1 -a)x+(b+1 -a)}}}
 {{{x+1}}}| {{{x^3+ax^2+bx-3}}}
........{{{x^3+x^2}}}
..............{{{ax^2-x^2}}}
...............{{{(1 -a)x^2+bx}}}
...............{{{(1 -a)x^2-(1 -a)x -3}}}
..........................{{{bx+(1 -a)x +(b+1 -a)}}}
................................................{{{-3-(b+1 -a) }}}
................................................. {{{-3-b-1+a  }}}
................................................  {{{ a-b -4}}} =>  reminder {{{ a-b -4 =-9}}}=> {{{a-b =-5}}}..........eq.2


go to

{{{b+a =3}}}..........eq.1, solve for {{{a}}}
{{{a=3-b}}}

go to

{{{a-b =-5}}}..........eq.2, substitute {{{a}}}
{{{3-b-b =-5}}}
{{{3-2b =-5}}}
{{{3+5 =2b}}}
{{{2b=8}}}
{{{b=4}}}

then

{{{a=3-b}}} => {{{a=3-4}}} => {{{a=-1}}}


your equation is

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


check reminders:

{{{x^3 - x^2 + 4 x - 3 = (x^2 + 4)*(x - 1) + 1}}}
{{{x^3 - x^2 + 4 x - 3 = (x^2 - 2 x + 6)*(x + 1) + (-9)}}}