complex functions

Hi,

Q1.
a) Find all complex-valued solutions to the equation z^3 - 2z^2 + 4z by hand. Write each one in the form a+bi.

b) Starting with the real-valued root, sketch all the complex-valued cuve roots of -27 in the complex plane. Against each, write its value in the form re^i(feta)


If there is anything unclear in the question, let me know, thanks.

Quote:

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Originally Posted by Moon Hoplite

Hi,

Q1.
a) Find all complex-valued solutions to the equation z^3 - 2z^2 + 4z by hand. Write each one in the form a+bi.

Mr F says: z(z^2 - 2z + 4). Factorise the quadratic by first completing the square.

b) Starting with the real-valued root, sketch all the complex-valued cuve roots of -27 in the complex plane. Against each, write its value in the form re^i(feta)
Mr F says: The real root cube root of -27 is -3. All the cube roots lie on a circle of radius 3 and are equally spaced by angle 2 pi/3 ....


If there is anything unclear in the question, let me know, thanks.

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..

in Q1, how does it mean by giving the answer in the form a+bi?

and in Q2., i still dont get it - is there any key words where i could research the topic based on this question?

Quote:

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Originally Posted by Moon Hoplite

in Q1, how does it mean by giving the answer in the form a+bi?

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Factor the quadratic and the answer should be clear.