Question 988995
When a polynomial {{{P(x)}}} is divided by {{{x-k}}} (where {{{k}}} is any real number) is {{{P(k)}}} .


1) {{{P(-2)=(-2)^3+m(-2)^2+n(-2)-5=-8+4m-2n-5=4m-2n-13}}}
If the remainder is {{{2}}} when {{{P(x)=x^3+mx^2+nx-5}}} is divided by {{{x+2=x-(-2))}}} ,
then {{{4m-2n-13=2}}}--->{{{4m-2n=15}}} .
{{{P(-3)=(-3)^3+m(-3)^2+n(-3)-5=-27+9m-3n-5=9m-3n-32}}}
If the remainder is {{{13}}} when {{{P(x)=x^3+mx^2+nx-5}}} is divided by {{{x+3=x-(-3))}}} ,
then {{{9m-3n-32=13}}}--->{{{9m-3n=45}}}--->{{{3m-n=15}}}--->{{{n=3m-15}}} .
{{{system(n=3m-15=45,4m-2n=15)}}}--->{{{system(n=3m-15=45,4m-2(3m-15)=15)}}}--->{{{system(n=3m-15=45,4m-6m+30=15)}}}
--->{{{system(n=3m-15=45,-2m+30=15)}}}--->{{{system(n=3m-15=45,-2m-30=15)}}}--->{{{system(n=3m-15=45,m=15/2)}}}
--->{{{system(n=3(15/2)-15=45,m=15/2)}}}--->{{{system(n=45/2-30/2=45,m=15/2)}}}--->{{{highlight(system(n=15/2,m=15/2))}}}


2) Similarly,
{{{P(1)=1^2+a*1-b=1+a-b=6}}}--->{{{a-b=6-1}}}--->{{{a-b=5}}} 
and {{{P(-2)=(-2)^2+a*(-2)-b=4-2a-b=3}}}--->{{{-2a-b=3-4}}}--->{{{-2a-b=-1}}}--->{{{2a+b=1}}}
{{{system(a-b=5,2a+b=1)}}}--->{{{system(a-b=5,a-b+2a+b=5+1)}}}--->{{{system(a-b=5,3a=6)}}}--->{{{system(a-b=5,a=2)}}}--->{{{system(2-b=5,a=2)}}}--->{{{system(-b=3,a=2)}}}--->{{{highlight(system(b=-3,a=2))}}}