Prove that (a + b + c)3 - a3 - b3 - c3 = 3(a + b)(b + c)(c + a). - Vidyadip

Question

Prove that (a + b + c)³ - a³ - b³ - c³ = 3(a + b)(b + c)(c + a).

✓

Answer

(a + b + c)³ = [(a + b + c)]³ = (a + b)³ + c³ + 3(a + b)c(a + b + c)
⇒ (a + b + c)³ = a³ + b³ + 3ab(a + b) + c³ + 3(a + b)c(a + b + c)
⇒ (a + b + c)³ - a³ + b³ - c³ = 3ab(a + b) + 3(a + b)c(a + b + c)
⇒ (a + b + c)³ - a³ + b³ - c³ = 3(a + b)[ab + ca + cb + c²]
⇒ (a + b + c)³ - a³ + b³ - c³ = 3(a + b)[a(b + c) + c(b + c)]
⇒ (a + b + c)³ - a³ + b³ - c³ = 3(a + b)(b + c)(a +c)

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