Question

factorise (x-y)³+(y-2)³+(2-x)³​

Answer 1

$\huge{\bigstar}{ \green{ \mathfrak{ \underline{ \underline{Question}}}}}{\bigstar}$

□☆□☆□☆□☆□☆□☆□☆□☆□☆□☆□☆□☆□☆□☆□☆□☆

$Factorise:\ (x-y)^3+(y-2)^3 + (2-x)^3$

□☆□☆□☆□☆□☆□☆□☆□☆□☆□☆□☆□☆□☆□☆□☆□☆

▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂

$\huge{\bigstar}{ \green{ \mathfrak{ \underline{ \underline{Answer}}}}}{\bigstar}$

$\boxed{\red{\bold{Let:}}}$

$\purple{x-y = a}$

$\purple{y-2 = b}$

$\purple{2-x = c}$

$\blue{\implies}a+b+c = x-y + y-2+2-x$

$\blue{\implies a+b+c = 0}$

$\boxed{\red{\bold{By\ identity:}}}$

$\blue{\implies a^3 + b^3 + c^3 -3abc = (a+b+c)(a^2 + b^2 +c^2 -2ab -2bc -2ca)}$

$\blue{\implies} a^3 + b^3 + c^3 -3abc = 0(a^2 +b^2 +c^2 - 2ab -2bc -2ca)$

${\blue{\implies}}a^3 + b^3 + c^3 -3abc = 0$

$\blue{\implies}a^3 +b^3 +c^3 = 3abc$

$\boxed{\red{\bold{Putting \ values \ of \ a, b\ and\ c:}}}$

$\green{\implies (x-y)^3 + (y-2)^3 + (2-x)^3 = 3(x-y)(y-2)(2-x)}$

$\boxed{\red{\bold{Hence:}}}$

$\green{\boxed{\implies{\boxed{(x-y)^3 + (y-2)^3 + (2-x)^3 = 3(x-y)(y-2)(2-x)}}}}$

▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂