Question 1186024



equal given factor to zero

{{{x^2-1=0}}}
{{{(x-1)(x+1)=0}}}
 => {{{x=1}}} or{{{ x=-1}}}

substitute the value of the factor as {{{x = 1}}} in the expression and equate to zero plus reminder as is a factor of the expression.
 
{{{x = 1}}}

{{{a*1^4+b*1^3-8*1^2+6*1-6 =0+2*1+1 }}}
{{{a + b - 8 = 3}}}
{{{a+b =3+8}}}
{{{a+b =11}}}
{{{a=11-b}}}....eq.1


 {{{x = -1}}}
{{{a*(-1)^4+b*(-1)^3-8*(-1)^2+6*(-1)-6 =0+2*(-1)+1 }}}
{{{a - b - 20 = -1}}}
{{{a = b +20  -1}}}
{{{a=b+19}}}....eq.2

from eq.1 and eq.2 we have

{{{11-b=b+19}}}
{{{11-19=b+b}}}
{{{2b=-8}}}
{{{b=-4}}}

go to eq.1 and substitute {{{b}}}
{{{a=11-(-4)}}}....eq.1
{{{a=11+4}}}
{{{a=15}}}

check
{{{(15x^4-4x^3-8x^2+6x-6 )/(x^2-1)}}}

{{{15 x^4 - 4 x^3 - 8 x^2 + 6 x - 6 = (15 x^2 - 4 x + 7)*(x^2 - 1) + (2 x + 1)}}}