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MATH / There is a polynomial P such that for every real number x,

Problem

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problem

There is a polynomial PP such that for every real number xx, x512+x256+1=(x2+x+1)P(x). x^{512} + x^{256} + 1 = (x^2 + x + 1) P(x). When PP is written in standard polynomial form, how many of its coefficients are nonzero?
Plain-text mathematical notation (without MathML)
There is a polynomial P such that for every real number x,
x⁵¹²+x²⁵⁶+1=(x²+x+1)P(x).When P is written in standard polynomial form, how many of its coefficients are nonzero?
Original LaTeX notation
There is a polynomial $P$ such that for every real number $x$,
\[
  x^{512} + x^{256} + 1 = (x^2 + x + 1) P(x).
\]When $P$ is written in standard polynomial form, how many of its coefficients are nonzero?

level

Level 5

type

Intermediate Algebra

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Source and history

task_id
a0770052-7a11-5da1-94e8-513c57fab8a3
task_key
intermediate~5falgebra--test--a0770052-7a11-5da1-94e8-513c57fab8a3
task_revision_id
1
config / split
intermediate_algebra / test

Official source

initial import