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yociyoci
 3 years ago
Solve x^4+x^3+x^2+x+1 = 0 by hand
yociyoci
 3 years ago
Solve x^4+x^3+x^2+x+1 = 0 by hand

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Mr.Math
 3 years ago
Best ResponseYou've already chosen the best response.3You're studying complex numbers I assume. I will give you a hint: \[x^4+x^3+x^2+x+1=\frac{x^51}{x1}.\] Solve it now by first finding the roots of unity of \(x^5=1\).

yociyoci
 3 years ago
Best ResponseYou've already chosen the best response.0Why is x^4+x^3+x^2+x+1 = (x^51)/(x1)

Mr.Math
 3 years ago
Best ResponseYou've already chosen the best response.3This come from the factorization \(x^51=(x1)(x^4+x^3+x^2+x+1)\). If you don't already know that. Then you can easily notice that \(1\) is a root of \(x^51\), and then you can use synthetic division to get the above factorization.

yociyoci
 3 years ago
Best ResponseYou've already chosen the best response.0but does that give you all of the roots?

Mr.Math
 3 years ago
Best ResponseYou've already chosen the best response.3\(x^51\) has all the roots of your problem and one more root which is \(x=1\), so you have to exclude it.

Mr.Math
 3 years ago
Best ResponseYou've already chosen the best response.3You know how to solve \(x^5=1\)?

yociyoci
 3 years ago
Best ResponseYou've already chosen the best response.01 and 4 complex solutions

Mr.Math
 3 years ago
Best ResponseYou've already chosen the best response.3Exactly. The four complex solutions are what you're looking for, as you can see from the factorization of \(x^51\).

yociyoci
 3 years ago
Best ResponseYou've already chosen the best response.0im not really sure about how to solve x^5=1

Mr.Math
 3 years ago
Best ResponseYou've already chosen the best response.3Have a look at this http://www.youtube.com/watch?v=9Z1uNz__2xA

moneybird
 3 years ago
Best ResponseYou've already chosen the best response.1Or First divide the entire equation by x^2 \[x^2+x+1+\frac{1}{x}+\frac{1}{x^2}=0 \implies x^2+\frac{1}{x^2}+x+\frac{1}{x}+1=0\] Let u = x+1/x \[u^2 = x^2+2+\frac{1}{x^2}\] The original equation now becomes: \[u^22+u+1 = 0 \implies u^2+u1=0\] Use quadratic equation then substitute u = x+1/x back and then solve the quadratic again.
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