Question
Evaluate: $(a^2 + b^2 + c^2 - ab - bc - ca)(a + b + c)$
✓
Answer
$(a^2 + b^2 + c^2 - ab - bc - ca)(a + b + c)$
$= a (a^2 + b^2 + c^2 - ab - bc - ca) + b (a^2 + b^2 + c^2 - ab - bc - ca) + c (a^2 + b^2 + c^2 - ab - bc - ca)$
$= a^3 + ab^2 + ac^2 - a^2b - abc - ca^2 +a^2b + b^3 + bc^2 - ab^2 - b^2c - abc + a^2c + b^2c + c^3 - abc - bc^2 - c^2a$
$= a^3 + b^3 + c^3 -a^2b + a^2b - ca^2 + a^2c + bc^2 - bc^2 - ab^2 + ab^2 - abc - abc - abc + ac^2 - ac^2 + b^2c - b^2c$
$= a^3 + b^3 + c^3 - 3abc$
$= a (a^2 + b^2 + c^2 - ab - bc - ca) + b (a^2 + b^2 + c^2 - ab - bc - ca) + c (a^2 + b^2 + c^2 - ab - bc - ca)$
$= a^3 + ab^2 + ac^2 - a^2b - abc - ca^2 +a^2b + b^3 + bc^2 - ab^2 - b^2c - abc + a^2c + b^2c + c^3 - abc - bc^2 - c^2a$
$= a^3 + b^3 + c^3 -a^2b + a^2b - ca^2 + a^2c + bc^2 - bc^2 - ab^2 + ab^2 - abc - abc - abc + ac^2 - ac^2 + b^2c - b^2c$
$= a^3 + b^3 + c^3 - 3abc$
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