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Step-by-step explanation:
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Step-by-step explanation:
(i) (2x + 3y)² = (2x)² + 2(2x)(3y) + (3y)²
= 4x² + 12xy + 9y²
(ii) (x - 2y)² = (x)² + 2(x)(2y) + (-2y)²
= x² - 4xy + 4y²
(iii) 4x² - 9y² = (2x)² - (3y)²
= (2x + 3y)(2x - 3y)
(iv) (x + 2y + 3z)² = (x)² + (2y)² + (3z)² + 2(x)(2y) + 2(2y)(3z) + 2(3z)(x)
= x² + 4y² + 9z² + 4xy + 12yz + 6zx
(v) 8x³ + 27y³ = (2x)³ + (3y)³
= (2x + 3y)[(2x)² - (2x)(3y) + (3y)²]
= (2x + 3y)(4x² - 6xy + 9y²)
(vi) 64x³ - 216y³ = (4x)³ - (6y)³
= (4x - 6y)[(4x)² + (4x)(6y) + (6y)²]
= (4x - 6y)(16x² + 24xy + 36y²)
(vii) (x + 5y)³ = (x)³ + (5y)³ + 3(x)(5y)(x + 5y)
= x³ + 125y³ + 3(x)²(5y) + 3(x)(5y)² [∵ 3ab(a + b) = 3a²b + 3ab²]
= x³ + 125y³ + 15x²y + 75xy²
(viii) (2x - y)³ = (2x)³ - (y)³ - 3(2x)²(y) - 3(2x)(y)²
= 8x³ - y³ - 12x²y - 6xy²
(ix) 8x³ + 216y³ + 125z³ - 180xyz = (2x + 6y + 5z)[(2x)² + (6y)² + (5z)² - (2x)(6y) - (6y)(5z) - (5z)(2x)
= (2x + 6y + 5z)(4x² + 36y² + 25z² - 12xy - 30yz - 10zx)
(x) a + b + c = 0
Then, a³ + b³ + c³ - 3abc = ?
a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ac)
= (0)(a² + b² + c² - ab - bc - ac) [∵ a + b + c = 0]
∴ a³ + b³ + c³ - 3abc = 0