let-P-n-x-1-2n-1-x-2n-1-1-prove-that-x-2-x-divide-P-n-

Question Number 73057 by mathmax by abdo last updated on 05/Nov/19

l e t P_{n} = {(x + 1)}^{2 n + 1} - x^{2 n + 1} - 1

p r o v e t h a t x^{2} + x d i v i d e \underset{n}{P}

Answered by MJS last updated on 06/Nov/19

x^{2} + x = 0 \Rightarrow x = - 1 \lor x = 0

P_{n} (0) = 1^{2 n + 1} - 0^{2 n + 1} - 1 = 0

P_{n} (- 1) = 0^{2 n + 1} - {(- 1)}^{2 n + 1} - 1 = 0

\Rightarrow \forall n \in N^{★} : (x^{2} + x) ∣ P_{n}

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