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We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we’ll assume that you are happy to receive all cookies from this website.Q:
If an irrational solution of $z^4+z^3+z^2+z+1 = 0$ is given, find two of its real roots.
I’m sorry for not explaining my concept well.
The equation I’ve got is: $$z^4+z^3+z^2+z+1 = 0$$
I know that the solutions of a polynomial are all real, and it must be an even function, so $z^4=-z^3-z^2-z+1$.
Then I’ve stuck, what can I do from now on?
Thanks!
A:
The complex solution to $z^4+z^3+z^2+z+1 = 0$ is, using $z=x+iy$,
$$
x=\frac{ -1\pm i\sqrt{3}}2,\quad
y=-\frac{1}{2},
$$
or
$$
x=\frac{ -1\mp i\sqrt{3}}2,\quad
y=\frac14-\frac{i}{2\sqrt3},
$$
where we have taken the positive sign in the second solution.
Therefore, the two real solutions of $z^4+z^3+z^2+z+1 = 0$ are
$$
x_1 = \frac{ -1+\sqrt{3}}2 = -\frac{\sqrt{3}-1}2,\qquad
x_2 = \frac{ -1-\sqrt{3}}2 = -\frac{\sqrt{3}+1}2.
$$
A:
We must solve
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