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Find the value

Question:

Find the value

$2 \sqrt{2} a^{3}+16 \sqrt{2} b^{3}+c^{3}-12 a b c$

 

Solution:

$=(\sqrt{2} a)^{3}+(2 \sqrt{2} b)^{3}+c^{3}-3 \times \sqrt{2} a \times 2 \sqrt{2} b \times c$

$=(\sqrt{2} a+2 \sqrt{2} b+c)\left((\sqrt{2} a)^{2}+(2 \sqrt{2} b)^{2}+c^{2}-(\sqrt{2} a)(2 \sqrt{2} b)-(2 \sqrt{2} b) c-(\sqrt{2} a) c\right)$

$=(\sqrt{2} a+2 \sqrt{2} b+c)\left(2 a^{2}+8 b^{2}+c^{2}-4 a b-2 \sqrt{2} b c-\sqrt{2} a c\right)$

$\therefore 2 \sqrt{2} a^{3}+16 \sqrt{2} b^{3}+c^{3}-12 a b c$

$=(\sqrt{2} a+2 \sqrt{2} b+c)\left(2 a^{2}+8 b^{2}+c^{2}-4 a b-2 \sqrt{2} b c-\sqrt{2} a c\right)$

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