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Problem If ∣ 1 1 1 a b c a 3 b 3 c 3 ∣ = ( a − b ) ( b − c ) ( c − a ) ( a + b + c ) \begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^3 & b^3 & c^3 \end{vmatrix} = (a-b)(b-c)(c-a)(a+b+c) 1 a a 3 1 b b 3 1 c c 3 = ( a − b ) ( b − c ) ( c − a ) ( a + b + c ) a , b a, b a , b c c c ∣ 1 1 1 ( x − a ) 2 ( x − b ) 2 ( x − c ) 2 ( x − b ) ( x − c ) ( x − c ) ( x − a ) ( x − a ) ( x − b ) ∣ \begin{vmatrix} 1 & 1 & 1 \\ (x-a)^2 & (x-b)^2 & (x-c)^2 \\ (x-b)(x-c) & (x-c)(x-a) & (x-a)(x-b) \end{vmatrix} 1 ( x − a ) 2 ( x − b ) ( x − c ) 1 ( x − b ) 2 ( x − c ) ( x − a ) 1 ( x − c ) 2 ( x − a ) ( x − b ) a + b + c = 0 a+b+c=0 a + b + c = 0 x = 1 3 ( a + b + c ) x = \frac{1}{3}(a+b+c) x = 3 1 ( a + b + c ) x = 1 2 ( a + b + c ) x = \frac{1}{2}(a+b+c) x = 2 1 ( a + b + c ) x = a + b + c x = a+b+c x = a + b + c Browse the problem library