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Let the equation x^(5) + x^(3) + x^(2) ...

Let the equation `x^(5) + x^(3) + x^(2) + 2 = 0` has roots `x_(1), x_(2), x_(3), x_(4) and x_(5),` then find the value of `(x_(1)^(2) - 1)(x_(2)^(2) - 1)(x_(3)^(2) - 1)(x_(4)^(2) - 1)(x_(5)^(2) - 1).`

Text Solution

Verified by Experts

The correct Answer is:
5
`x^(5) + x^(3) + x^(2) + 2 = (x -x_(1)) (x -x_(2))(x -x_(3))(x -x_(4))(x -x_(5))`
Putting x = 1, we get,
`5 = (1 -x_(1)) (1 -x_(2))(1 -x_(3))(1 -x_(4))(1 -x_(5))`
Putting x = - 1, we get,
`1 = (-1 -x_(1)) (-1 -x_(2))(-1 -x_(3))(-1 -x_(4))(-1 -x_(5))`
Multiplying, we get,
`5 = (x_(1)^(2) - 1)(x_(2)^(2) - 1)(x_(3)^(2) - 1)(x_(4)^(2) - 1)(x_(5)^(2) - 1)`
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