2. The expression (a - b)³+(b-c)³ +(c-a)³ can be factorized as?
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Answer:
3 [ ab(b - a) + bc(c - b) + ca(c - a) ]
Step-by-step explanation:
Given : (a - b)³ + (b - c)³ + (c - a)³
Identity : (x - y)³ = x³ - y³ - 3x²y + 3xy²
→ [ (a)³ - (b)³ - 3(a)²(b) + 3(a)(b)² ] + [ (b)³ - (c)³ - 3(b)²(c) + 3(b)(c)² ] + [ (c)³ - (a)³ - 3(c)²(a) + 3(c)(a)² ]
→[ a³ - b³ - 3a²b + 3ab² ] + [ b³ - c³ - 3b²c + 3bc² ] + [ c³ - a³ - 3c²a + 3ca² ]
→ a³ - b³ - 3a²b + 3ab² + b³ - c³ - 3b²c + 3bc² + c³ - a³ - 3c²a + 3ca²
→ a³ - a³ - b³ + b³ - c³ + c³ - 3a²b - 3b²c - 3c²a + 3ab² + 3bc² + 3ca²
→ - 3a²b - 3b²c - 3c²a + 3ab² + 3bc² + 3ca²
→ - 3a²b + 3ab² - 3b²c + 3bc² - 3c²a + 3ca²
→ 3 [ ab(b - a) + bc(c - b) + ca(c - a) ]
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