Question

Verify:

x³+ y³+ z³- 3xyz =(x+y+z) [(x-y)² + (y-z)² +(z-x)²] and a factor of 64x³+125y³ -64³+ 240.

Answer 1

x³ + y³ + z³ - 3xyz = (x + y + z){x² + y² + z² - xy - yz - zx }
=1/2 (x + y + z){2x² + 2y² + 2z² -2xy - 2yz - 2zx }
= 1/2 (x + y + z){( x² -2xy + y²) + (y² - 2yz + z²) +(z² - 2zx + x²) }
= 1/2 (x + y + z){(x - y)² + (y - z)² + (z - x)²}

Hence,verified

Now,

64x³ + 125y³ - 64z³ +240xyz
= (4x)³ + (5y)³ +(-4z)³ -3(4x)(5y)(-4z)
This is just like above Identity , use that .

=1/2 (4x + 5y -4z ){(4x -5y)² + (5y+4z)² +(4x + 4z)² }

Hence, (4x + 5y -4z ) and {(4x -5y)² + (5y+4z)² +(4x + 4z)² } are the factors of given expression.