Q. 233.8( 4 Votes )

# If x + y + z = 0,

Answer :

Given: x + y + z = 0

To Prove: Taking LHS, Expanding along the first row, we get

= xa{(za)(ya) – (xc)(xb)} – (yb){(yc)(ya) – (zb)(xb)} + (zc){(yc)(xc) – (zb)(za)}

= xa{a2yz – x2bc} – yb{y2ac – b2xz} + zc{c2xy – z2ab}

= a3xyz – x3abc – y3abc + b3xyz + c3xyz – z3abc

= xyz(a3 + b3 + c3) – abc(x3 + y3 + z3)

It is given that x + y + z = 0

x3 + y3 + z3 = 3xyz

= xyz(a3 + b3 + c3) – abc (3xyz)

= xyz(a3 + b3 + c3 – 3abc) Hence Proved

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