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The determinant with rows (x², (y+z)², yz), (y², (x+z)², zx), (z², (x+y)², xy) is divisible by which of the following?
- x² + y² + z²
- x - y
- x - y - z
- x + y + z
Correct answer: x + y + z
Solution
Setting x + y + z = 0 makes the determinant zero (the three rows become linearly dependent), so (x+y+z) is a factor. Also setting x = y makes two rows identical, so (x-y) is a factor. The determinant is divisible by both (x+y+z) and (x-y) (and similarly (y-z) and (z-x)).
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