Question 1177973
 Find the value of {{{k}}} if 


{{{(x^3-x^2-5x-2)(x^4+x^3+kx^2-5x+2)=x^7-4x^5-14x^4-5x^3+19x^2-4}}}....multiply left side


{{{x^7+ kx^5 - k x^4 - 5kx^3 - 2 kx^2  - 6 x^5 - 12 x^4 + 5 x^3 + 23 x^2 - 4=x^7-4x^5-14x^4-5x^3+19x^2-4 }}}...............simplify


{{{cross(x^7)+ kx^5 - k x^4 - 5kx^3 - 2 kx^2  - 6 x^5 - 12 x^4 + 5 x^3 + 23 x^2 - 4=cross(x^7)-4x^5-14x^4-5x^3+19x^2-4}}}


{{{kx^5 - k x^4 - 5kx^3 - 2 kx^2  - 6 x^5 - 12 x^4 + 5 x^3 + 23 x^2 - 4+4x^5+14x^4+5x^3-19x^2+4=0}}}


{{{k x^5 - k x^4 - 5 k x^3 - 2 k x^2 - 2 x^5 + 2 x^4 + 10 x^3 + 4 x^2 = 0}}}


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


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


that will be true if {{{k-2=0}}}


{{{k=2}}}



check:

{{{(x^3-x^2-5x-2)(x^4+x^3+kx^2-5x+2)=x^7-4x^5-14x^4-5x^3+19x^2-4}}}

{{{(x^3-x^2-5x-2)(x^4+x^3+2x^2-5x+2)=x^7-4x^5-14x^4-5x^3+19x^2-4}}}

{{{x^7 - 4x^5 - 14x^4 - 5x^3 + 19x^2 - 4=x^7-4x^5-14x^4-5x^3+19x^2-4}}} which is true