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Polynomial and Its General Form

If P=n n2-1...

Question

If $P = n (n^{2} - 1^{2}) (n^{2} - 2^{2}) (n^{2} - 3^{2}) . . . (n^{2} - r^{2}), n > r, n ϵ N,$ then P is divisible by

A

(2 r + 2)!

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B

(2 r - 1)!

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C

(2 r + 1)!

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D

none of these

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Solution

The correct options are
B $(2 r - 1)!$
D $(2 r + 1)!$
$P = n (n^{2} - 1) (n^{2} - 2^{2}) . . . (n^{2} - r^{2})$
$P = n (n - 1) (n + 1) (n - 2) (n + 2) . . . (n - r) (n + r)$
Rearranging the terms we get,
$P = (n - r) (n - r + 1) . . . (n + r - 1) (n + r)$
This is a product of $2 r + 1$ consecutive terms. Hence it must be divisible by $(2 r + 1)!$
Hence, options B and C are correct.

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