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The determinant with rows [a², a²-(b-c)², bc], [b², b²-(c-a)², ca], [c², c²-(a-b)², ab] is divisible by which of the following?

1. a + b + c
2. (a + b)(b + c)(c + a)
3. a² + b² + c²
4. (a - b)(b - c)(c - a)

Correct answer: (a - b)(b - c)(c - a)

Solution

Setting a = b makes the determinant zero, so (a-b) divides it; by symmetry (b-c) and (c-a) also divide it, giving the factor (a-b)(b-c)(c-a).

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