Question

prove that (a-b)^3+(b-c)^3+(c-a)^3=3(a-b)(b-c)(c-a)​

Answer 1

we know that

$x^{3} + {y}^{3} + {z}^{3} - 3xyz = (x + y + z)( {x}^{2} + {y}^{2} + {z}^{2} - xy - yz - zx)$

by equation

if x+y+z=0

${x}^{3} + {y}^{3} + {z}^{3} = 3xyz$

we know that

(a-b)+(b-c)+(c-a)=0

=>

$(a - b) ^{3} + {(b - c)}^{3} + {(c - a)}^{3} = 3(a - b)(b - c)(c - a)$

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