Question

please solve it fast ​

Answer 1

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm :\longmapsto\:\dfrac{ {( {x}^{2} - {y}^{2})}^{3} + {( {y}^{2} - {z}^{2})}^{3} + {( {z}^{2} - {x}^{2})}^{3} }{ {(x - y)}^{3} + {(y - z)}^{3} + {(z - x)}^{3} }$

Consider, Numerator

$\rm :\longmapsto\: {( {x}^{2} - {y}^{2})}^{3} + {( {y}^{2} - {z}^{2})}^{3} + {( {z}^{2} - {x}^{2})}^{3}$

Let assume that

$\rm :\longmapsto\:a = {x}^{2} - {y}^{2}$

$\rm :\longmapsto\:b = {y}^{2} - {z}^{2}$

$\rm :\longmapsto\:c = {z}^{2} - {x}^{2}$

On adding above 3 equations,

$\rm :\longmapsto\:a + b + c = {x}^{2} - {y}^{2} + {y}^{2} - {z}^{2} + {z}^{2} - {x}^{2}$

$\rm :\longmapsto\:a + b + c = 0$

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

On substitute the values of a, b and c,

$\rm :\longmapsto\: {( {x}^{2} - {y}^{2})}^{3} + {( {y}^{2} - {z}^{2})}^{3} + {( {z}^{2} - {x}^{2})}^{3}$

$\rm \:  =  \:  \: \: 3{( {x}^{2} - {y}^{2}}){( {y}^{2} - {z}^{2})} {( {z}^{2} - {x}^{2})}$

Consider, denominator

$\rm :\longmapsto\: {(x - y)}^{3} + {(y - z)}^{3} + {(z - x)}^{3}$

Let assume that

$\rm :\longmapsto\:a = x - y$

$\rm :\longmapsto\:b = y - z$

$\rm :\longmapsto\:c = z - x$

On adding above 3 equations,

$\rm :\longmapsto\:a + b + c = x - y + y - z + z - x$

$\rm :\longmapsto\:a + b + c = 0$

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

On substituting the values of a, b, c

$\rm :\longmapsto\: {(x - y)}^{3} + {(y - z)}^{3} + {(z - x)}^{3}$

$\rm \:  =  \:  \: 3(x - y)(y - z)(z - x)$

Consider,

$\rm :\longmapsto\:\dfrac{ {( {x}^{2} - {y}^{2})}^{3} + {( {y}^{2} - {z}^{2})}^{3} + {( {z}^{2} - {x}^{2})}^{3} }{ {(x - y)}^{3} + {(y - z)}^{3} + {(z - x)}^{3} }$

$\rm \:  =  \:  \: \dfrac{3( {x}^{2} - {y}^{2} )( {y}^{2} - {z}^{2} )( {z}^{2} - {x}^{2} )}{3(x - y)(y - z)(z - x)}$

$\rm \:  =  \:  \: \dfrac{(x - y)(x + y)(y + z)(y - z)(z - x)(z + x)}{(x - y)(y - z)(z - x)}$

$\rm \:  =  \:  \: (x + y)(y + z)(z + x)$

More to know

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

a² - b² = (a + b)(a - b)

(a + b)² = (a - b)² + 4ab

(a - b)² = (a + b)² - 4ab

(a + b)² + (a - b)² = 2(a² + b²)

(a + b)³ = a³ + b³ + 3ab(a + b)

(a - b)³ = a³ - b³ - 3ab(a - b)