Show that the solutions for $1 + z^{4} + z^{3} + 2 z^{2} = 0$ $1 + z^4 + z^3 + 2 z^2 =0$ obey the condition $| z | = 1$ $absz = 1$ ?

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Show that the solutions for $1 + z^{4} + z^{3} + 2 z^{2} = 0$ $1 + z^4 + z^3 + 2 z^2 =0$ obey the condition $| z | = 1$ $absz = 1$ ?

2 Answers

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Answer 1 · 2017-12-21T16:36:35

$| z | = 1$ $|z| = 1$

Explanation:

$This is a symmetric equation. Those can be solved with the$ $"This is a symmetric equation. Those can be solved with the"$
$substitution t = z + 1/z :$ $"substitution t = z + 1/z : "$
$\Rightarrow t^{2} + t = 0 .$ $=> t^2 + t = 0.$
$\Rightarrow t = 0 or t = - 1$ $=> t = 0 or t = -1$
$\Rightarrow z^{2} + 1 = 0 or z^{2} + z + 1 = 0$ $=> z^2 + 1 = 0 or z^2 + z + 1 = 0$
$\Rightarrow z = \pm i or z = \frac{- 1 \pm \sqrt{3} i}{2}$ $=> z = pm i or z = (-1 pm sqrt(3) i)/2$
$\Rightarrow | z | = 1$ $=> |z| = 1$

Answer 2 · 2017-12-21T17:16:24

$| z | = 1$ $absz = 1$

Explanation:

$1 + z^{4} + z^{3} + 2 z^{2} = (1 + z^{2}) (1 + z + z^{2}) = 0$ $1 + z^4 + z^3 + 2 z^2 = (1+z^2)(1+z+z^2) = 0$

then $| z | = 1$ $absz = 1$

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Show that the solutions for $1 + z^{4} + z^{3} + 2 z^{2} = 0$ $1 + z^4 + z^3 + 2 z^2 =0$ obey the condition $| z | = 1$ $absz = 1$ ?

2 Answers

Explanation:

Explanation:

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Show that the solutions for 1+z4+z3+2z2=01 + z^4 + z^3 + 2 z^2 =0 obey the condition |z|=1absz = 1 ?

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Explanation:

Explanation:

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Show that the solutions for $1 + z^{4} + z^{3} + 2 z^{2} = 0$ $1 + z^4 + z^3 + 2 z^2 =0$ obey the condition $| z | = 1$ $absz = 1$ ?