I am having trouble with the following.

Factorise $x^{111}+9x^{74}+27x^{37}+27$ in irreducible factors in $\mathbb{Z}[x]$.

I did not find it to be an Eisentein polynomial and trying to find zeros by hand is a lot of work, so is a linear substitution of $x$ by $cx+b$. Also, reducing the polynomial modulo a prime number (for example 3) did not help.

  • 6
    $\begingroup$ Compute $2\times 37$ and $3\times 37$ and see if that lights the bulb. $\endgroup$ – B. Goddard Mar 25 at 18:11
  • $\begingroup$ As suggested by B. Goddard, with wild looking polynomials it is often useful to look for a pattern in the exponents on the variable. The hint by cansomeonehelpmeout puts the icing on the cake $\endgroup$ – Chris Leary Mar 25 at 18:34
  • $\begingroup$ $\left(x^{37}+3\right)^3$ $\endgroup$ – David G. Stork Mar 25 at 19:13


Make the substitution $u=x^{37}$

For completion. After the hint, you can write your expression like $$\color{green}{1}\cdot 3^0\cdot u^3+\color{green}{3}\cdot 3^1\cdot u^2+\color{green}{3}\cdot 3^2\cdot u+\color{green}{1}\cdot 3^3\\(u+3)^3=(x^{37}+3)^3$$


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