What are the solutions to the equation z^3+z^2+z=0 ?

The solutions to the equation z^3+z^2+z=0 are z=0,z=-1/2+i(sqrt(3))/2 ,z=-1/2-i(sqrt(3))/2

Find the zeros of z^3+z^2+z=0

The zeros of z^3+z^2+z=0 are z=0,z=-1/2+i(sqrt(3))/2 ,z=-1/2-i(sqrt(3))/2

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z^{3} + z^{2} + z = 0

z = 0, z = - \frac{1}{2} + i \frac{3}{2}, z = - \frac{1}{2} - i \frac{3}{2}

Solution steps

z^{3} + z^{2} + z = 0

Factor

z^{3} + z^{2} + z : z (z^{2} + z + 1)

z (z^{2} + z + 1) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

z = 0 or z^{2} + z + 1 = 0

Solve

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

The solutions are

z = 0, z = - \frac{1}{2} + i \frac{3}{2}, z = - \frac{1}{2} - i \frac{3}{2}

2 x^{3} + 2 x^{2} + 2 x + 2 = 0

7 x^{3} - 47 x^{2} + 121 x - 65 = 0

x^{2} - 4 = x^{3} - 2 x^{2}

(x^{2} - 2)^{2} - 4 (x^{2} - 2) + 4 = 0

x^{4} + x^{2} = \frac{4}{5}

What are the solutions to the equation z^3+z^2+z=0 ?
The solutions to the equation z^3+z^2+z=0 are z=0,z=-1/2+i(sqrt(3))/2 ,z=-1/2-i(sqrt(3))/2
Find the zeros of z^3+z^2+z=0
The zeros of z^3+z^2+z=0 are z=0,z=-1/2+i(sqrt(3))/2 ,z=-1/2-i(sqrt(3))/2