When is $n^2 x^{2n+3} - 25n x^{n+1} + 150x^7$ divisible by $x^2-1$?

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The polynomial

\[n^2 x^{2n+3} - 25n x^{n+1} + 150x^7\]

has \(x^2 - 1\) as a factor

1. for no values of \(n\),

2. for \(n=10\) only,

3. for \(n=15\) only,

4. for \(n=10\) and \(n=15\) only.

Given that \(x^2-1\) is a factor, what linear factors of the polynomial do we know?

Can we now use the factor theorem on the polynomial twice?

We may need to be more careful with one factor than the other…