Expand to seek trouble
∣ ∣ ∣ ∣ ∣ ∣ ( a − x ) 2 ( b − x ) 2 ( c − x ) 2 ( a − y ) 2 ( b − y ) 2 ( c − y ) 2 ( a − z ) 2 ( b − z ) 2 ( c − z ) 2 ∣ ∣ ∣ ∣ ∣ ∣ = − 8 3 5 1
If x , y , z are roots of the equation 8 X 3 − 6 2 X 2 + 4 3 X − 7 = 0 , and they satisfy the determinant above, where a , b and c are distinct numbers, find the value of ∣ ( a − b ) ( b − c ) ( c − a ) ∣ .
The answer is 2.00.
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1 solution
Lovely problem involving an interesting factorization and good use of the Vandermonde Determinant! (+1 and liked)
I think you should ask for the absolute value of ( a − b ) ( b − c ) ( c − a ) ; the product could be − 2 as well. Also, it is not necessary to require that a , b and c are distinct.
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Thank you sir , edited!
Just one small doubt , how could one get − 2 . Could u elaborate plz?
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It depends on the order in which you write the roots: Let a = 1 , b = 0 , c = 2 , x = 7 , y = 1 / 4 , z = 1 / 2 , for example.
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@Otto Bretscher – Okay, thank u once again.
The roots of the equation are 7 , 2 1 , 4 1
And the determinant can be rewritten as
∣ ∣ ∣ ∣ ∣ ∣ ∣ a 2 b 2 c 2 − 2 a − 2 b − 2 c 1 1 1 ∣ ∣ ∣ ∣ ∣ ∣ ∣ × ∣ ∣ ∣ ∣ ∣ ∣ ∣ 1 1 1 x y z x 2 y 2 z 2 ∣ ∣ ∣ ∣ ∣ ∣ ∣
(Row to Row multiplication)
Well known determinants
This evaluates to 2 ( a − b ) ( b − c ) ( c − a ) ( x − y ) ( y − z ) ( z − x )
hence,
( a − b ) ( b − c ) ( c − a ) = 2