$x^{15 } +1$ as a product of four factors.

Question

How can we write $$x^{15 } +1$$ as a product of four factors.

I can only discern one thing about this, that $x^{3} +1$,$x^{5 } +1$ and $x^{15 } +1$ are going to be the factors. But I am unable to move ahead.

You can begin by noticing that the fifteenth roots of unity contains the third roots of unity. Performing long division on $(x^{15}+1)/(x^5 +1)$ leads to a simpler expression. — Henricus V., Dec 26 '16 at 21:57

score 3 · Answer 1 · answered Dec 26 '16 at 21:20

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There just so happens to be this wonderful online calculator known as WolframAlpha, and upon putting your problem in, we find

$$x^{15}+1=(x+1)(x^2-x+1)(x^4-x^3+x^2-x+1)(x^8+x^7-x^5-x^4-x^3+x+1)$$

Particularly, notice that

$$x^3+1=(x+1)(x^2-x+1)$$

$$x^5+1=(x+1)(x^4-x^3+x^2-x+1)$$

answered Dec 26 '16 at 21:20

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I'm not up-voting this due to your sarcasm in the beginning of your answer. – Dec 26 '16 at 21:26
@floorcat and what sarcasm? WolframAlpha is most definitely a wonderful online calculator. – Simply Beautiful Art Dec 26 '16 at 21:28
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It's comforting to read about your complacence with answers when you hit your cap. – Dec 26 '16 at 21:30
@floorcat I am mildly addicted as the user ranked third this week and fourth this month. :-) – Simply Beautiful Art Dec 26 '16 at 21:33
Could you please stop it? – Catalin Zara Dec 26 '16 at 21:34
@CatalinZara Stop what? I'm sorry if I'm self-indulging, I've never hit this much rep in one day before. – Simply Beautiful Art Dec 26 '16 at 21:36
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@SimpleArt: I thought this site was about helping others, not about self-indulging or bickering with each other. – Catalin Zara Dec 26 '16 at 21:38
@CatalinZara yes it is, I was just overly excited. Sorry, and Happy Holidays. – Simply Beautiful Art Dec 27 '16 at 01:54
hey @floorcat, sarcasm, really?? It is okay that you feel something rude here but I do not think that mentioning that "I do not upvote because of your sarcasm" is compleately wrong. There is no harm in making your answer funny. – Vidyanshu Mishra Dec 27 '16 at 14:05
@CatalinZara, read the above message it was for you too. – Vidyanshu Mishra Dec 27 '16 at 14:06
@floorcat, and I shall upvote it too. why?? for sarcasm. – Vidyanshu Mishra Dec 27 '16 at 14:07

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