Question 1154891

Find a polynomial of degree {{{3}}} such that when divided by {{{x^2-5x}}} has a remainder of {{{6x-15}}} and when divided by {{{x^2-5x+8}}} has a remainder of {{{2x-7}}}.



{{{f(x)/g(x)=q(x)+reminder}}}

{{{f(x)=q(x)*g(x)+reminder}}}


if given {{{g(x)= x^2-5x}}}  and {{{reminder =6x-15}}}, we have

{{{f(x)=q(x)*(x^2-5x )+6x-15}}}.......eq.1


and  if given {{{g(x)= x^2-5x+8}}}  and {{{reminder =2x-7}}}, we have

{{{f(x)=q(x)*(x^2-5x+8)+2x-7}}}.......eq.2


from eq.1 and eq.2 we have


{{{q(x)*(x^2-5x )+6x-15=q(x)*(x^2-5x+8)+2x-7}}}

{{{q(x)*(x^2-5x )-q(x)*(x^2-5x+8)=2x-6x+15-7}}}.........solve for {{{q(x)}}}

{{{q(x)(x^2-5x -(x^2-5x+8))=-4x+8}}}

{{{q(x)(x^2-5x -x^2+5x-8)=-4x+8}}}.......simplify

{{{q(x)(-8)=-4x+8}}}

{{{q(x)=-4x/(-8)+8/(-8)}}}

{{{q(x)=x/2-1}}}-> your quotient


now find {{{f(x)}}}

{{{f(x)=q(x)*g(x)+reminder}}}

{{{f(x)=(x/2-1)*(x^2-5x )+6x-15}}}.......eq.1

{{{f(x)=x^3/2-5x^2/2-x^2 +5x+6x-15}}}

{{{f(x)=x^3/2 - 7x^2/2 +11x - 15}}}=> your polynomial of degree {{{3}}}