How Do You Solve and Plot g(z) = 0 for Complex Roots on an Argand Diagram?

[noahsdev]

Homework Statement

Let f(z) = z³-8 and g(z) = f(z-1). This information applies to questions 1-5.
1. Express g(z) in the form g(z) = z³+az² +bz + c
2. Hence, solve g(z) = 0. Plot solutions on an Argand diagram.

Homework Equations

Factorisation
i²=-1

The Attempt at a Solution

I have done question 1.
g(z) = z³- 3z² +3z - 9.
With question 2 I think z = 3 or z = √3i or -√3i. But (assuming I am correct) I am not sure what I do next. I am sure it is the simplest solution in the world but I have looked at it for so long I have confused myself.

Also as a side note, when plotting complex numbers is it a point or a line from the origin?

 

[HallsofIvy]

Those are not quite correct- you don't have the real part of the last two. If you write u= z- 1, then g(z)= f(z- 1)= f(u)= u^3- 8= 0. So, in polar form, u^3= 8= 2^3= 2^3(e^{2\pi ni}) so u= 2e^{2\pi ni/3}= 2(cos(2npi/3)+ isin(2n\pi/3)) with n= 0, 1, and 2.

Those are the three points equally spaced around the circle with radius 2, centered on 0 with one (n= 0) the point 2 itself. In "a+ bi" terms, they are u= 2 (n= 0), u= -2+ (\sqrt{3})i (n= 1), and u= -2- (\sqrt{3})i (n= 2).

Of course, z= u+ 1.

Also as a side note, when plotting complex numbers is it a point or a line from the origin?

In other words, you are trying to do a problem involving an "Argand diagram" without knowing what an "Argand diagram" is! Didn't it occur to you look it up? Each complex number, a+ bi, is plotted as the point (a, b) on an Argand diagram.

 

[Curious3141]

Those are not quite correct- you don't have the real part of the last two.

He does. It just cancels out when you add the 1.

If you write u= z- 1, then g(z)= f(z- 1)= f(u)= u^3- 8= 0. So, in polar form, u^3= 8= 2^3= 2^3(e^{2\pi ni}) so u= 2e^{2\pi ni/3}= 2(cos(2npi/3)+ isin(2n\pi/3)) with n= 0, 1, and 2.

Those are the three points equally spaced around the circle with radius 2, centered on 0 with one (n= 0) the point 2 itself. In "a+ bi" terms, they are u= 2 (n= 0), u= -2+ (\sqrt{3})i (n= 1), and u= -2- (\sqrt{3})i (n= 2).

The complex roots are wrong. The real part should be -1 for each, since they start out as -1/2 before you multiply by 2.

Of course, z= u+ 1.

And so the real parts cancel out, giving what the threadstarter got.

EDIT: Thought better of it, while the question is ambiguous, it seems the De Moivre's solution is probably the insight they were looking for.

In other words, you are trying to do a problem involving an "Argand diagram" without knowing what an "Argand diagram" is! Didn't it occur to you look it up? Each complex number, a+ bi, is plotted as the point (a, b) on an Argand diagram.

In most of the places I've looked, the complex number is represented either by a line segment or an arrow pointing away from the origin. Not just a point. Of course, the line segment or arrow terminates at (a,b).