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What is the LCM of (x 2 - y 2 - z 2 - 2yz), (x 2 - y 2 + z 2 + 2xz) and (x 2 + y 2 - z 2 - 2xy)?

Q: What is the LCM of (x 2 - y 2 - z 2 - 2yz), (x 2 - y 2 + z 2 + 2xz) and (x 2 + y 2 - z 2 - 2xy)?

x2 - y2 - z2 - 2yz = x2 - (y + z)2 = (x+ y + z)(x - y - z) x2 - y2 + z2 + 2yz = (x + z)2 - y2 = (x + y + z) (x + y - z) x2 + y2 - z2 - 2xy = (x - y)2 - z2 = (x - y + z) (x - y - z) ∴ LCM = (x + y + z) (x - y - z) (x - y + z)

(x + y + z) (x + y - z) (x - y + z)
(x + y + z) (x - y - z) (x - y + z)
(x + y + z) (x + y - z) (x - y - z)
(x + y - z) (x - y - z) (x - y + z)

Correct Answer: (x + y + z) (x - y - z) (x - y + z)

Explanation:

x² - y² - z² - 2yz = x² - (y + z)² = (x+ y + z)(x - y - z)
x² - y² + z² + 2yz = (x + z)² - y² = (x + y + z) (x + y - z)
x² + y² - z² - 2xy = (x - y)² - z² = (x - y + z) (x - y - z)
∴ LCM = (x + y + z) (x - y - z) (x - y + z)

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