Question

Verify : x³ + y³ = (x + y)(x² - xy + y²)

And on this basis simplify :

27y³ + 125z³​

Answer 1

☆Solution :

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• Verify : x³ + y³ = (x + y)(x² - xy + y²)

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☞︎︎︎ x³ + y³ = x(x² - xy + y²) + y(x² - xy + y²)

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• Distributive Property

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⟹ x³ + y³ = ( x³ - x²y + xy²) + (yx² - xy² + y³)

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• Cancel the like terms with opposite sign..

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$\rm = ( x³ \cancel{ - x²y} + xy²) + ( \cancel{x²y} - xy² + y³)$

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$\rm = ( x³ { } + \cancel{ xy²}) + ( {} \cancel{ - xy²} + y³)$

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☞︎︎︎ x³ + y³ = x³ + y³

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☆ Now :

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i) 27y³ + 125z³

< Simplify = Factorise >

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☞︎︎︎ Using Identity-

x³ + y³ = ( x³ - x²y + xy²) + (yx² - xy² + y³)

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⟹ 27y³ + 125z³ = (3y)³ + (5z)³

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• a = 3y

• b = 5z

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⟹ (3y)³ + (5z)³ = (3y + 5z) [(3y)² - (3y)(5z) + (5z)²]

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= (3y + 5z)(9y² - 15yz + 25z²)

• hence simplified