Question

Resolve into factors: 27x3 + y3 + z3 - 9xyz

Answer 1

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27x³ + y³ + z³ - 9xyz

= ( 3x )³ + ( y )³ + ( z )³ - 3 × 3x × y × z

= 1 / 2 ( 3x + y + z ) { ( 3x - y )² + ( y - z )² + ( z - 3x )² }

[ • We know ,
✬ a³ + b³ + c³- 3abc = 1 / 2 ( a + b + c ) { ( a - b )² + ( b - c )² + ( c - a )² } ]

[ • Here , a = 3x , b = y , c = z ]

= 1 / 2 ( 3x + y + z ) { 9x² - 6xy + y² + y² - 2yz + z² + z² - 6xz + x² }

= 1 / 2 ( 3x + y + z ) { 10x² + 2y² + 2z² - 6xy - 2yz - 6xz }

= 1 / 2 × 2 ( 3x + y + z ) { 5x² + y² + z² - 3xy - yz - 3xz }

= ( 3x + y + z ) ( 5x² + y² + z² - 3xy - yz - 3xz ) [ ★ Required answer ]

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Answer 2

a^3+b^3+c^3+3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)
required answer is (3x+y+z)(9x^2+y^2+z^2-3xy-yz-3xz)