Question

if x+y+z=0, show that x^3 + y^3 + z^3

step by step​

Answer 1

Step-by-step explanation:

$\huge\mathfrak{given}$

$\textbf{x + y + z = 0}$

$\bold{\implies x + y = - z}$

$\bold\red{cubing \: on \: both \: sides}$

$\bold{ {(x + y)}^{3} = {( - z)}^{3} }$

$\bold{\implies {x}^{3} + {y}^{3} + {3x}^{2} y + {3xy}^{2} = { - z}^{3} }$

$\bold{\implies {x}^{3} + {y}^{3} + 3xy(x + y) = { - z}^{3} }$

$\bold{\implies {x}^{3} + {y}^{3} + 3xy( - z) = { - z}^{3} }$

$\bold{\implies {x}^{3} + {y}^{3} - 3xyz = { - z}^{3} }$

$\bold{\implies {x}^{3} + {y}^{3} + {z}^{3} = 3xyz}$