Question
Multiply $x^2 + 4y^2 + z^2 + 2xy + xz - 2yz by(-z + x - 2y).$
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Answer
$(x^2 + 4y^2 + z^2 + 2xy + xz - 2yz)(-z + x - 2y)$
$= x^2(-z + x - 2y) + 4y^2(-z + x - 2y) + z^2(-z + x - 2y)$$ + 2xy(-z + x - 2y) + xz(-z + x - 2y) - 2yz(-z + x - 2y)$
$= x^2z + x^3 - 2x^2y - 4y^2z + 4xy^2 - 8y^3 - z^3 + xz^2 - 2yz^2 $$- 2xyz + 2x^2y - 4xy^2 - xz^2 + x^2z - 2xyz + 2yz^2 - 2xyz + 4y^2z$
$= (-x^2z + x^2z ) + x^3 + (-2x^2y + 2x^2y) + (-4y^2z + 4y^2z) + $$(4xy^2- 4xy^2)- 8y^3 - z^3 + (xz^2 - xz^2) $$+ (-2yz^2+ 2yz^2) + (-2xyz - 2xyz - 2xyz)$
$= x^3 - 8y^3 - z^3 - 6xyz$
Alternate Answer:
Now, $(x - 2y - z)(x^2 + 4y^2 + z^2 + 2xy + xz - 2yz)$
$= (x - 2y - z)[(x)^2 + (-2y)^2 + (-z)^2 - (x)(-2y) - (-2y)(-z) - (x)(-z)]$
$= (x^3) + (-2y)^3 + (-z)^3 - 3(x)(-2y)(-z)$
[Using identity, $a^3 + b^3+ c^3 - 3abc$
$= (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)]$
$= x^3 - 8y^3 - z^3 - 6xyz$
$= x^2(-z + x - 2y) + 4y^2(-z + x - 2y) + z^2(-z + x - 2y)$$ + 2xy(-z + x - 2y) + xz(-z + x - 2y) - 2yz(-z + x - 2y)$
$= x^2z + x^3 - 2x^2y - 4y^2z + 4xy^2 - 8y^3 - z^3 + xz^2 - 2yz^2 $$- 2xyz + 2x^2y - 4xy^2 - xz^2 + x^2z - 2xyz + 2yz^2 - 2xyz + 4y^2z$
$= (-x^2z + x^2z ) + x^3 + (-2x^2y + 2x^2y) + (-4y^2z + 4y^2z) + $$(4xy^2- 4xy^2)- 8y^3 - z^3 + (xz^2 - xz^2) $$+ (-2yz^2+ 2yz^2) + (-2xyz - 2xyz - 2xyz)$
$= x^3 - 8y^3 - z^3 - 6xyz$
Alternate Answer:
Now, $(x - 2y - z)(x^2 + 4y^2 + z^2 + 2xy + xz - 2yz)$
$= (x - 2y - z)[(x)^2 + (-2y)^2 + (-z)^2 - (x)(-2y) - (-2y)(-z) - (x)(-z)]$
$= (x^3) + (-2y)^3 + (-z)^3 - 3(x)(-2y)(-z)$
[Using identity, $a^3 + b^3+ c^3 - 3abc$
$= (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)]$
$= x^3 - 8y^3 - z^3 - 6xyz$
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