What is the remainder if $p^{11} + p^{9} + p^{7} + p^{5} + p^{3} + p^{2}$ divided by p + 1

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Solution

Let $P^{11} + P^{7} + P^{3} + P^{2} = f (P)$
Using remainder theorem, get
Remainder $= f (P) = f (- 1) = (- 1)^{11} + (- 1)^{9} + (- 1)^{7} + (- 1)^{5} + (- 1)^{3} + (- 1)^{2} = - 4$

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