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Omni-MATH / We call a two-variable polynomial P(x,y) [i]secretly one-variable,[/i] if there exist polynomials Q(x) and R(x,y) such that deg(Q) ge 2 and P(x,y)=Q(R(x,y)) (e.g. x²+1 and x²y²+1 a…

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We call a two-variable polynomial P(x,y)P(x, y) [i]secretly one-variable,[/i] if there exist polynomials Q(x)Q(x) and R(x,y)R(x, y) such that $\deg(Q) \ge 2$ and P(x,y)=Q(R(x,y))P(x, y) = Q(R(x, y)) (e.g. x2+1x^2 + 1 and x2y2+1x^2y^2 +1 are [i]secretly one-variable[/i], but xy+1xy + 1 is not). Prove or disprove the following statement: If P(x,y)P(x, y) is a polynomial such that both P(x,y)P(x, y) and P(x,y)+1P(x, y) + 1 can be written as the product of two non-constant polynomials, then PP is [i]secretly one-variable[/i].
Plain-text mathematical notation (without MathML)
We call a two-variable polynomial P(x,y) [i]secretly one-variable,[/i] if there exist polynomials Q(x) and R(x,y) such that $\deg(Q) \ge 2$ and P(x,y)=Q(R(x,y)) (e.g. x²+1 and x²y²+1 are [i]secretly one-variable[/i], but xy+1 is not).

Prove or disprove the following statement: If P(x,y) is a polynomial such that both P(x,y) and P(x,y)+1 can be written as the product of two non-constant polynomials, then P is [i]secretly one-variable[/i].
Original LaTeX notation
We call a two-variable polynomial $P(x, y)$ [i]secretly one-variable,[/i] if there exist polynomials $Q(x)$ and $R(x, y)$ such that $\deg(Q) \ge 2$ and $P(x, y) = Q(R(x, y))$ (e.g. $x^2 + 1$ and $x^2y^2 +1$ are [i]secretly one-variable[/i], but $xy + 1$ is not).

Prove or disprove the following statement: If $P(x, y)$ is a polynomial such that both $P(x, y)$ and $P(x, y) + 1$ can be written as the product of two non-constant polynomials, then $P$ is [i]secretly one-variable[/i].

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Source and history

task_id
7ba22670-8fc9-5322-ae5b-ff962d5aaddb
task_key
test--7ba22670-8fc9-5322-ae5b-ff962d5aaddb
task_revision_id
1
config / split
/ test

Official source

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