Question

${x - y}^{3) + {y - z}^{3} + {z - x}^{3} ka man hai$

Answer 1

✍✍HERE IS YOUR ANSWER...⤵⤵
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$\underline{SOLUTION}$ ➡

$We \: \: know \: \: that ,\\ \\ (a + b + c) {}^{3} = a {}^{3} + b {}^{3} + c {}^{3} + 3(a + b)(b + c)(c + a) \\ \\ = > a {}^{3} + b {}^{3} + c {}^{3} = (a + b + c) {}^{3} - 3( a + b)(b + c)(c + a)$

$Let, \\ \\ a = x - y \\ \\ b = y - z \\ \\ c = z - x$

$So, \\ \\ a + b + c = (x - y) + (y - z) +( z - x )\\ \\ = > a + b + c = 0 \\ \\ \\ a + b =( x - y )+ (y - z )\\ \\ = > a + b = x - z \\ \\ \\ b + c =( y - z) + (z - x) \\ \\ = > b + c = y - x \\ \\ \\ c + a = (z - x) + (x - y) \\ \\ = > c + a = z - y$

$\: Now ,\\ \\a {}^{3} + b {}^{3} + c {}^{3} = (a + b + c) {}^{3} - 3( a + b)(b + c)(c + a) \\ \\ = > (x - y) {}^{3} + (y - z) {}^{3} + (z - x) {}^{3} = (0) {}^{3} - 3(x - z)(y - x)(z - y) \\ \\ = > (x - y) {}^{3} + (y - z) {}^{3} + (z - x) {}^{3} = 0 - 3(x - z)(y - x)(z - y) \\ \\ = > (x - y) {}^{3} + (y - z) {}^{3} + (z - x) {}^{3} = - 3(x - z)(y - x)(z - y) \\ \\ = > (x - y) {}^{3} + (y - z) {}^{3} + (z - x) {}^{3} = 3(x - y)(y - z)(z - x)$

$\underline{ANSWER}$ ➡

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

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