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{2 π}{9}) + i sin (\frac{2 π}{9}), x = cos (\frac{8 π}{9}) + i sin (\frac{8 π}{9}), x = cos (\frac{14 π}{9}) + i sin (\frac{14 π}{9}), x = cos (\frac{2 π}{9}) - i sin (\frac{2 π}{9}), x = cos (\frac{4 π}{9}) + i sin (\frac{4 π}{9}), x = cos (\frac{10 π}{9}) + i sin (\frac{10 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{2 π}{9}) + i sin (\frac{2 π}{9}), x = cos (\frac{8 π}{9}) + i sin (\frac{8 π}{9}), x = cos (\frac{14 π}{9}) + i sin (\frac{14 π}{9}), x = cos (\frac{2 π}{9}) - i sin (\frac{2 π}{9}), x = cos (\frac{4 π}{9}) + i sin (\frac{4 π}{9}), x = cos (\frac{10 π}{9}) + i sin (\frac{10 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{2 π}{9}) + i sin (\frac{2 π}{9}), x = cos (\frac{8 π}{9}) + i sin (\frac{8 π}{9}), x = cos (\frac{14 π}{9}) + i sin (\frac{14 π}{9}), x = cos (\frac{2 π}{9}) - i sin (\frac{2 π}{9}), x = cos (\frac{4 π}{9}) + i sin (\frac{4 π}{9}), x = cos (\frac{10 π}{9}) + i sin (\frac{10 π}{9})

x^{4} + 16 = 10 x^{2}

4 x^{3} - 26 x = 0

x^{4} - 6 x^{3} + 11 x^{2} - 6 x = 0

x^{4} + x^{3} - 6 x^{2} = 0

- x^{4} + x^{2} + 12 = 0

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