Question

Arătați că (2x+1)^3-8x-4= (2x-1)(2x+1)(2x+3)

Answer 1

cazul 1) daca x= -1/2 => 0 +4-4=0 (adevarat)
cazul 2) x diferit de -1/2 =>
=> (2x+1)^3-8x-4= (2x-1)(2x+1)(2x+3) <=>
<=>(2x+1)^3 -4*(2x+1)=(2x-1)(2x+1)(2x+3) <=>
<=>(2x+1)*((2x+1)^2-4)=(2x-1)(2x+1)(2x+3)<=>
<=>(2x+1)^2-4=(2x-1)(2x+3)<=>
<=>4x^2+1+4x-4=4x^2+6x-2x-3<=>
<=>4x^2+4x-3=4x^2+4x-3 <=>0=0 (adevarat)

Answer 2

(2x+1)*(2x+1)*(2x+1)-8x-4 = (2x+1)^2*(2x+1)-4*(2x+1)  =

[(2x+1)-2^2]*(2x+1) =  (2x+1-2)*(2x+1+2)*(2x+1)  =  (2x-1)(2x+1)(2x+3) .