Complex Numbers Help

[darkphoenix]

Got a question, couldn't quite figure it out.

1. Write down a polynomial of degree 3, whose coefficients are all real, that has 4i and 2 as two of its zeros.

I tried using simultaneous equations, but it didn't quite work.

[mandy]

Seeing as the coefficients are all real, wouldn't you use the Conjugate Root Theorem? Then, you would have three factors that you could expand and get the polynomial from.

i.e.

...I hope that's right.

[Martoman]

Seeing as the coefficients are all real, wouldn't you use the Conjugate Root Theorem? Then, you would have three factors that you could expand and get the polynomial from.

i.e.

...I hope that's right.

You are correct. You beat me to it

[darkphoenix]

Omg LOL

Thanks alot mandy. 

Dam i didnt think it was that simple and i started subbing stuff in..ah never mind.

[mandy]

Omg LOL

Thanks alot mandy. 

Dam i didnt think it was that simple and i started subbing stuff in..ah never mind.

Haha, glad I could help

[darkphoenix]

Ok i have another question:

2. Factorise over C

[kamil9876]

solve

then the polynomial is where are the roots.

[Martoman]

solve

then the polynomial is where are the roots.

Alternatively



Then simply factorise this as a perfect cube. However, this would still facilitate solving for , essentially what Kamil is talking about.

[darkphoenix]

Hm ive managed to get as one of the factors but cant seem to find the other quadratic factor

[Martoman]

Use demoivre's theorm.



Then

Solutions occur by equating sides, so

Then

Now solutions occur at:
Shove in values of k until it repeats. I would shove in Once you done that convert to cartesian. Then you have your roots.

[Martoman]

Hm ive managed to get as one of the factors but cant seem to find the other quadratic factor

Alternatively then just divide however you do, normally or synthetically.

[darkphoenix]

Alright got it.

Cheers dude.