Question 1047655


Determine the value of {{{k}}} in {{{f(x) = 5x^3+kx^2-4x+4}}} so that {{{x+2}}} is a factor 


That {{{(x+2)}}} is a factor of {{{f(x)=5x^3+kx^2-4x+4}}}  means exactly that
{{{f(-2)=0}}}.

That is you want to solve the equation:

{{{5(-2)^3+k(-2)^2-4(-2)+4 =0}}}  for {{{k}}}

{{{5(-8)+4k+8+4 =0}}}

{{{-40+4k+12 =0}}}

{{{-28+4k =0}}}

{{{4k =28}}}

{{{k =28/4}}}

{{{k =7}}}

so, we have

{{{f(x)=5x^3+7x^2-4x+4}}}

check if {{{x+2}}} is factor

{{{f(x)=5x^3+7x^2-4x+4}}}....write {{{7x^2}}} as {{{10x^2-3x^2}}} and {{{-4x}}} as  {{{-6x+2x}}}

{{{f(x)=5x^3+10x^2-3x^2-6x+2x+4}}}...group

{{{f(x)=(5x^3+10x^2)-(3x^2+6x)+(2x+4)}}}

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

{{{f(x)=(x+2) (5x^2-3x+2)}}}......so,{{{x+2}}} is factor