Răspuns:
Explicație pas cu pas:
x o y = xy + 4x + 4y + 12 ; (∀) x,y ∈ R
a) x o y = (x+4)(y+4) - 4 ?
x o y = xy + 4x + 4y + 12 = x(y+4) + 4(y+3) =
= x(y+4) + 4(y+3) +4 - 4 = x(y+4) + 4(y+3+1) -4 =
= x(y+4) + 4(y+4) -4 = (x+4)(y+4) - 4
b) x o (-4) = x·(-4) + 4x + 4·(-4) + 12 = -4x+4x-16+12 = -4 ; (∀) x ∈ R
c) '' o '' asociativa ?
<=> (x o y ) o z = x o (y o z)
(x o y ) o z = (xy + 4x + 4y + 12) o z =
= (xy + 4x + 4y + 12)·z + 4·(xy + 4x + 4y + 12) +4·z +12 =
= xyz + 4xz+4yz+12z+4xy+16x+16y+48+4z+12 =
= xyz + 4xz+4yz+4xy+16x+16y+16z+60
x o (y o z) = x o (yz+4y+4z+12) =
= x·(yz+4y+4z+12)+4·x+4·(yz+4y+4z+12)+12 =
= xyz + 4xy + 4xz + 12x + 4x + 4yz + 16y + 16z + 48 +12 =
= xyz + 4xy + 4xz + 4yz + 16x + 16y + 16z + 60 =>
(x o y ) o z = x o (y o z) => legea '' o '' este asociativa
#copaceibrainly
