Question

solution pls
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Answer 1

Given Question

Factorize the following

$\rm :\longmapsto\: {a}^{2}(b - c) + {b}^{2}(c - a) + {c}^{2}(a - b)$

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

Given expression is

$\rm :\longmapsto\: {a}^{2}(b - c) + {b}^{2}(c - a) + {c}^{2}(a - b)$

$\rm \:  =  \: {a}^{2}(b - c) + {cb}^{2} - {ab}^{2} + {ac}^{2} - {bc}^{2}$

can be re-arranged as

$\rm \:  =  \: {a}^{2}(b - c) + {cb}^{2} - {bc}^{2} + {ac}^{2} - {ab}^{2}$

$\rm \:  =  \: {a}^{2}(b - c) + cb(b - c) + a( {c}^{2} - {b}^{2} )$

$\rm \:  =  \: {a}^{2}(b - c) + cb(b - c) - a( {b}^{2} - {c}^{2} )$

$\rm \:  =  \: {a}^{2}(b - c) + cb(b - c) - a(b - c)(b + c)$

$\rm \:  =  \: (b - c)\bigg[{a}^{2} + cb - a(b + c)\bigg]$

$\rm \:  =  \: (b - c)\bigg[{a}^{2} + cb - ab - a c\bigg]$

$\rm \:  =  \: (b - c)\bigg[{a}^{2} - ab + bc - a c\bigg]$

$\rm \:  =  \: (b - c)\bigg[ a(a - b) - c(a - b)\bigg]$

$\rm \:  =  \: (b - c)\bigg[ (a - b)(a - c)\bigg]$

$\rm \:  =  \: (b - c) (a - b)(a - c)$

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More Identities 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)