What are the solutions to the equation x^{10}=-x ?

The solutions to the equation x^{10}=-x are

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x^{10}=-x

x^{10} = - x

x = 0, x = - 1, x = \frac{1}{2} + i \frac{3}{2}, x = \frac{1}{2} - i \frac{3}{2}, x = cos (\frac{π}{9}) + i sin (\frac{π}{9}), x = cos (\frac{7 π}{9}) + i sin (\frac{7 π}{9}), x = cos (\frac{13 π}{9}) + i sin (\frac{13 π}{9}), x = cos (\frac{π}{9}) - i sin (\frac{π}{9}), x = cos (\frac{5 π}{9}) + i sin (\frac{5 π}{9}), x = cos (\frac{11 π}{9}) + i sin (\frac{11 π}{9})

Solution steps

x^{10} = - x

Move

x

to the left side

x^{10} + x = 0

Factor

x^{10} + x : x (x + 1) (x^{2} - x + 1) (x^{6} - x^{3} + 1)

x (x + 1) (x^{2} - x + 1) (x^{6} - x^{3} + 1) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

x = 0 or x + 1 = 0 or x^{2} - x + 1 = 0 or x^{6} - x^{3} + 1 = 0

Solve

x + 1 = 0 : x = - 1

Solve

x^{2} - x + 1 = 0 : x = \frac{1}{2} + i \frac{3}{2}, x = \frac{1}{2} - i \frac{3}{2}

Solve

x^{6} - x^{3} + 1 = 0 : x = cos (\frac{π}{9}) + i sin (\frac{π}{9}), x = cos (\frac{7 π}{9}) + i sin (\frac{7 π}{9}), x = cos (\frac{13 π}{9}) + i sin (\frac{13 π}{9}), x = cos (\frac{π}{9}) - i sin (\frac{π}{9}), x = cos (\frac{5 π}{9}) + i sin (\frac{5 π}{9}), x = cos (\frac{11 π}{9}) + i sin (\frac{11 π}{9})

The solutions are

x = 0, x = - 1, x = \frac{1}{2} + i \frac{3}{2}, x = \frac{1}{2} - i \frac{3}{2}, x = cos (\frac{π}{9}) + i sin (\frac{π}{9}), x = cos (\frac{7 π}{9}) + i sin (\frac{7 π}{9}), x = cos (\frac{13 π}{9}) + i sin (\frac{13 π}{9}), x = cos (\frac{π}{9}) - i sin (\frac{π}{9}), x = cos (\frac{5 π}{9}) + i sin (\frac{5 π}{9}), x = cos (\frac{11 π}{9}) + i sin (\frac{11 π}{9})

- x^{3} + 9 x^{2} - 24 x + 20 = 0

(x^{2} - 8)^{2} - 2 (x^{2} - 8) + 1 = 0

- 4 x^{5} = 256 x^{2}

8 = a^{3}

0 = 2 x^{3} + 8 x^{2} + 14 x + 24

What are the solutions to the equation x^{10}=-x ?
The solutions to the equation x^{10}=-x are
Find the zeros of x^{10}=-x
The zeros of x^{10}=-x are