Question
Prove that $(a + b + c)^3 - a^3 - b^3 - c^3 = 3(a + b)(b + c)(c + a).$
✓
Answer
$(a + b + c)^3 = [(a + b + c)]^3 = (a + b)^3 + c^3 + 3(a + b)c(a + b + c)$
$\Rightarrow (a + b + c)^3 = a^3 + b^3 + 3ab(a + b) + c^3 + 3(a + b)c(a + b + c)$
$\Rightarrow (a + b + c)^3 - a^3 + b^3 - c^3 = 3ab(a + b) + 3(a + b)c(a + b + c)$
$\Rightarrow (a + b + c)^3 - a^3 + b^3 - c^3 = 3(a + b)[ab + ca + cb + c^2]$
$\Rightarrow (a + b + c)^3 - a^3 + b^3 - c^3 = 3(a + b)[a(b + c) + c(b + c)]$
$\Rightarrow (a + b + c)^3 - a^3 + b^3 - c^3 = 3(a + b)(b + c)(a +c)$
$\Rightarrow (a + b + c)^3 = a^3 + b^3 + 3ab(a + b) + c^3 + 3(a + b)c(a + b + c)$
$\Rightarrow (a + b + c)^3 - a^3 + b^3 - c^3 = 3ab(a + b) + 3(a + b)c(a + b + c)$
$\Rightarrow (a + b + c)^3 - a^3 + b^3 - c^3 = 3(a + b)[ab + ca + cb + c^2]$
$\Rightarrow (a + b + c)^3 - a^3 + b^3 - c^3 = 3(a + b)[a(b + c) + c(b + c)]$
$\Rightarrow (a + b + c)^3 - a^3 + b^3 - c^3 = 3(a + b)(b + c)(a +c)$
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