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

let P_n =(x+1)^(2n+1) −x^(2n+1) −1 prove that x^2 +x divide P_n

letPn=(x+1)2n+1−x2n+1−1provethatx2+xdividePn

Answered by MJS last updated on 06/Nov/19

x^2 +x=0 ⇒ x=−1∨x=0 P_n (0)=1^(2n+1) −0^(2n+1) −1=0 P_n (−1)=0^(2n+1) −(−1)^(2n+1) −1=0 ⇒ ∀n∈N^★ : (x^2 +x)∣P_n

x2+x=0⇒x=−1∨x=0Pn(0)=12n+1−02n+1−1=0Pn(−1)=02n+1−(−1)2n+1−1=0⇒∀n∈N★:(x2+x)∣Pn

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