Factoring a Septic Polynomial | Algebra Challenge

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It would be easier to apply your (very useful btw) trick to subtract and then add x to power of whatever minus three a second time:
x⁴ + x² + 1 = x⁴ - x + x² + x + 1 = x(x³ - 1) + (x² + x + 1) = (x² - x + 1) (x² + x + 1)

MrGeorge
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I was worried your account might have been hacked at 07:48 until 08:40 where I verified it is the classic and typical you! 😂

skalf
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Again, like in the earlier 5th and 11th degree examples: just start with the unit circle! That's much more straightforward and understandable. (Here you will quickly see that the cube root of 1 and its conjugate are both solutions of the original polynomial, and then you know you can divide by (x-r)(x-r*) = x^2+x+1, if we call this cube root and its conjugate r and r*.)

JosBergervoet