Question

Aratati ca (x-5)³ -x+5=(x-4)(x-5)(x-6),pentru orice numar real x.
Rezolvare completa va rogg.

Answer 1

(x-5)(x²-10x+25)-x+5=(x²-5x-4x+20)(x-6)
x³-10x²+25x-5x²+50x-125-x+5=x³-6x²-5x²+30x-4x²+24x+20x-120
x³-15x²+74x-120=x³-15x²+74x-120   ⇒ca pentru orice numar real x va exista egalitate

Answer 2

Descopunem in factori expresia:
$(x-5)^3-x+5=(x-5)^3-(x-5)=(x-5)(x-5)^2-1=\\ =(x-5)(x-5-1)(x-5+1)=(x-5)(x-6)(x-4)=\\=(x-4)(x-5)(x-6)$
Am scos factorul comun pe (x-5) si am ramas cu (x-5)(x-5) la a doua - 1, apoi am aplicat formula a la a doua - b la a doua si am obtinut 
(x-5)(x-5-1)(x-5+1) care imi da exact concluzia:
(x-4)(x-5)(x-6).
Success!