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For any non-n...

Question

For any non-negative integers $m, n, p$ , prove that the polynomial $x^{3 m} + x^{3 n + 1} + x^{3 p + 2}$ has the factor $x^{2} + x + 1$

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Solution

For sake of simplicity let us assume $m > n > p$
Then, $x^{3 m} + x^{3 n + 1} + x^{3 p + 2}$ can be written as
$x^{3 p + (3 m - 3 p)} + x^{3 p + (3 n + 1 - 3 p)} + x^{3 p + 2}$ ( remember $3 m > 3 n > 3 p, 3 p$ is the smallest among them ).
Take $x^{3 p}$ as common,
$x^{3 p} (1 + x + x^{2})$ ( other terms ).
So, $(1 + x + x^{2})$ will always be a factor of polynomial of the form $x^{3 m} + x^{3 n + 1} + x^{3 p + 2} .$
Hence, the answer is $x^{3 m} + x^{3 n + 1} + x^{3 p + 2} .$

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