Question

Aratati ca valoarea expresiei E(X) = x³+2x²-3x supra x²+3x este un numar natural ,pentru orice X,natural nenul

Answer 1

x³+2x²-3x = x(x² + 2x - 3) = x(x² + 3x -x - 3) = x[x(x + 3) - 1(x + 3)] =
= x(x+3)(x-1)

x² + 3x = x(x+3)

$\frac{x(x+3)(x-1)}{x(x+3)}$

(se simplifica x si x+3)

E(x) = x - 1 ∈ N

Answer 2

$\frac{x^3+2x^2-3x}{x^2+3x}= \frac{x(x^2+2x-3)}{x(x+3)}= \frac{x^2+2x-3}{x+3}$ =$\frac{(x+3)(x-1)}{x+3}=x-1$ ∈ N ,∀ x ∈ N*