Question

A(x)=$x^{4}$+2$x^{3}$-$x^{x^{2} }$-2n

aratati ca se divide cu 8 oricare ar fi x numar natural diferit de 0

Answer 1

[tex]A(x)=x^4+2x^3-x^2-2x\\ A(x)=x^3(x+2)-x(x+2)\\ A(x)=(x+2)(x^3-x)\\ A(x)=x(x+2)(x^2-1)\\ A(x)=x(x+2)(x-1)(x+1)\\ Cei\ patru\ termeni\ sunt\ consecutivi\Rightarrow A\vdots 4\ (1)\\ Dar\ si\ x(x+1)\ sunt\ termeni\ consecutivi\Rightarrow A\vdots 2\ (2)\\ Din\ (1)\ si\ (2)\Rightarrow A\vdots 8\\[/tex]