Question
Factorize the following expressions:
$a^3+8 b^3+64 c^3-24 a b c$
$a^3+8 b^3+64 c^3-24 a b c$
✓
Answer
$a^3 + 8b^3 + 64c^3 - 24abc$
$= (a)^3 + (2b)^3 + (4c)^3 - 3 \times 2b \times 4c$
$= (a + 2b + 4c)(a^2 + (2b)^2 + (4c)^2 - a \times 2b - 2b \times 4c - 4c \times a)$
$\big[\because$ $a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca$)$\big]$
$= (a + 2b + 4c)(a^2 + 4b^2+ 16c^2 - 2ab - 8bc - 4ac)$
$\therefore$ $a^3 + 8b^3 + 64c^3 - 24abc$
$= (a + 2b + 4c)(a^2 + 4b^2 + 16c^2 - 2ab - 8bc - 4ac)$
$= (a)^3 + (2b)^3 + (4c)^3 - 3 \times 2b \times 4c$
$= (a + 2b + 4c)(a^2 + (2b)^2 + (4c)^2 - a \times 2b - 2b \times 4c - 4c \times a)$
$\big[\because$ $a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca$)$\big]$
$= (a + 2b + 4c)(a^2 + 4b^2+ 16c^2 - 2ab - 8bc - 4ac)$
$\therefore$ $a^3 + 8b^3 + 64c^3 - 24abc$
$= (a + 2b + 4c)(a^2 + 4b^2 + 16c^2 - 2ab - 8bc - 4ac)$
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