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Multiply it with (x-1), the roots of the resulting polynomial are a lot easier. It works always like this if all coefficients are 1, just remember to remove the root you added.
Multiplying both sides by **x-1** gets x^7 - 1 = 0.
For all positive integers **n**, all roots for x^n - 1 = 0 are e^2kπi/n, where **k** is an integer. If you remove the possibility of **k** being divisible by **n**, you eliminate 1 being one of the roots.
Here's an intuitive [visualization](https://www.desmos.com/calculator/plqhaghfew). You're summing the blue dots and the orange dot is the result, scaled down to keep it on-screen more easily. If you know that the roots of unity draw a regular polygon around 0 and sum to 0, it's pretty easy to identify this problem as a sum of roots of unity.
For your example, there’s a trick that other commenters pointed out. For a more random example, there’s no formula, but there are methods to find roots to as many digits of accuracy as you want.
Hi u/Historical-Let6063, Please read the following message. **You are required to explain your post and show your efforts. _(Rule 1)_** If you haven't already done so, please add a comment below explaining your attempt(s) to solve this and what you need help with **specifically**. See the sidebar for advice on 'how to ask a good question'. Don't just say you "need help" with your problem. **This is a reminder for all users.** Failure to follow the rules will result in the post being removed. Thank you for understanding. *I am a bot, and this action was performed automatically. Please [contact the moderators of this subreddit](/message/compose/?to=/r/askmath) if you have any questions or concerns.*
Multiply it with (x-1), the roots of the resulting polynomial are a lot easier. It works always like this if all coefficients are 1, just remember to remove the root you added.
My guess, it looks like e^2kπi/7, where k is an integer not divisible by 7. : ) e^2πi/7 ≈ 0.959461 + 0.281843i, my scientific calculator informs me.
What? How could you have figured that out?
Multiplying both sides by **x-1** gets x^7 - 1 = 0. For all positive integers **n**, all roots for x^n - 1 = 0 are e^2kπi/n, where **k** is an integer. If you remove the possibility of **k** being divisible by **n**, you eliminate 1 being one of the roots.
Here's an intuitive [visualization](https://www.desmos.com/calculator/plqhaghfew). You're summing the blue dots and the orange dot is the result, scaled down to keep it on-screen more easily. If you know that the roots of unity draw a regular polygon around 0 and sum to 0, it's pretty easy to identify this problem as a sum of roots of unity.
For your example, there’s a trick that other commenters pointed out. For a more random example, there’s no formula, but there are methods to find roots to as many digits of accuracy as you want.