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Reciprocal polynomials. A polynomial \(P = \sum_{k=0}^{n} a_k X^k \in \mathbb{R}[X]\) of degree \(n \geq 1\) is called reciprocal if \(a_k = a_{n-k}\) for every \(k \in \{0, \dots, n\}\) (its sequence of coefficients is a palindrome).
  1. Show that \(P\) is reciprocal if and only if \(X^n P(1/X) = P(X)\) for every \(X \neq 0\).
  2. Show that if \(P\) is reciprocal and \(\alpha \in \mathbb{C} \setminus \{0\}\) is a root of \(P\), then \(1/\alpha\) is also a root of \(P\), with the same multiplicity.
  3. Factor \(P = X^4 - X^3 - X + 1\) in \(\mathbb{R}[X]\).
Hint for (3). Verify \(P\) is reciprocal, then divide by \(X^2\) and substitute \(Y = X + 1/X\).
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