Simplify: $(2 x + y + 4 z) (4 x^{2} + y^{2} + 16 z^{2} - 2 x y - 4 y z - 8 z x)$ .

8 x^{3} + y^{3} + 64 z^{3} - 24 x y z

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8 x^{3} - y^{3} + 64 z^{3} - 24 x y z

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8 x^{3} + y^{3} - 64 z^{3} - 24 x y z

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None of these

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Solution

The correct option is B $8 x^{3} + y^{3} + 64 z^{3} - 24 x y z$
We know that one of the identity is $(a + b + c) (a^{2} + b^{2} + c^{2} - a b - b c - c a) = a^{3} + b^{3} + c^{3} - 3 a b c$ .

Now, consider the given expression $(2 x + y + 4 z) (4 x^{2} + y^{2} + 16 z^{2} - 2 x y - 4 y z - 8 z x)$ and compare it with the above identity as shown below:

$(2 x + y + 4 z) (4 x^{2} + y^{2} + 16 z^{2} - 2 x y - 4 y z - 8 z x) = (2 x + y + 4 z) [(2 x)^{2} + y^{2} + (4 z)^{2} - (2 x \times y) - (y \times 4 z) - (2 x \times 4 z)] = (2 x)^{3} + y^{3} + (4 z)^{3} - (3 \times 2 x \times y \times 4 z) = 8 x^{3} + y^{3} + 64 z^{3} - 24 x y z .$

Hence, $(2 x + y + 4 z) (4 x^{2} + y^{2} + 16 z^{2} - 2 x y - 4 y z - 8 z x) = 8 x^{3} + y^{3} + 64 z^{3} - 24 x y z$ .

Therefore, option $A$ is correct.

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