z = 1 + 0i
So
|rz| = 1 and az = 0 radians.
which allows you to write z = rz*cos(az) + i*sin(az)
Then, if y = z1/3 then
|y| = |z1/3| = |11/3| = 1
and
ay is the angle in [0, 360) such that 3*ay = 0 mod(2*pi)
that is, ay = 0, 2pi/3 and 4pi/3
And therefore,
Root 1 = cos(0) + i*sin(0)
Root 2 = cos(2pi/3) + i*sin(2pi/3) and
Root 3 = cos(4pi/3) + i*sin(4pi/3).
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two complex
If you graph it, you will find there is only one real root at approx x = -0.868145. The other 2 roots are complex numbers. I don't remember the steps, to get that.
(1+i)3 = 1 + 3i - 3 - i = -2 + 2i This is a complex number, and therefore cannot be plotted on a Cartesian plane.
The four roots are:1 + 2i, 1 - 2i, 3i and -3i.
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(x - 4)(3x^3 + 4) The complex roots are the cube root of negative one third times two to the two thirds power and negative two to the two thirds power divided by the cube root of three
It has two complex roots.
two complex
There are no real root. The complex roots are: [-5 +/- sqrt(-3)] / 2
3x^2 + 8x + 4 = (x+2) (3x+2) x = -2 x = -2/3 So there are no complex roots, they are real. You can test this by b^2 - 4ac if greater than 0, it is real if equal, there will be 2 identical roots. if less than 0 you get imaginary roots.
You can find the roots with the quadratic equation (a = 1, b = 3, c = -5).
If you graph it, you will find there is only one real root at approx x = -0.868145. The other 2 roots are complex numbers. I don't remember the steps, to get that.
(1+i)3 = 1 + 3i - 3 - i = -2 + 2i This is a complex number, and therefore cannot be plotted on a Cartesian plane.
The four roots are:1 + 2i, 1 - 2i, 3i and -3i.
9 cube !
This quadratic equation has no real roots because its discriminant is less than zero.