Solving Polynomials with Complex Coefficients

[||spoon||]

Hey,

When solving polynomials over c that have complex coefficients such as:
z^3+(5i-4)z^2+(3-20i)z+15i
what is the easiest way to find your first factor? My textbook says to use the factor theorem, if you agree is there a quicker way to find a factor than by trialing the constants factors?

Also, for an equation such as:
Z^2+(3-4i)z-12i=0
or:
z^2-(3+2i)z+6i=0
the solutions are z=-3 z=4i and z=3 z=2i respectively. Is it just cpincidence that these solutions "correspond" to the same numerical values as is in the complex coefficient or is there some form of rule for this type of equation to quickly solve them at a glance?

Thanks

Btw i did not know wether this should go into honewirk or not. I chose not because it isnr homework and i can solve these with ease anyway lol. Just curious about different methods and speed :), cheers

-||spoon||

 

[John Creighto]

Hey,

When solving polynomials over c that have complex coefficients such as:
z^3+(5i-4)z^2+(3-20i)z+15i
what is the easiest way to find your first factor? My textbook says to use the factor theorem, if you agree is there a quicker way to find a factor than by trialing the constants factors?

Also, for an equation such as:
Z^2+(3-4i)z-12i=0
or:
z^2-(3+2i)z+6i=0
the solutions are z=-3 z=-4i and z=3 z=2i respectively. Is it just cpincidence that these solutions "correspond" to the same numerical values as is in the complex coefficient or is there some form of rule for this type of equation to quickly solve them at a glance?

Thanks

Btw i did not know wether this should go into honewirk or not. I chose not because it isnr homework and i can solve these with ease anyway lol. Just curious about different methods and speed :), cheers

-||spoon||

I'm not sure. I wonder if there is an equivalent to the rational root theorem for the types of problems you are trying to solve. I think there exists a general closed form formula for the roots of a cubic. For a more general treatment of algebraic solutions to polynomials pick up a book on group theory or modern algebra.

 

[||spoon||]

no I meant the factor theorem... If you have a polynomial P(z) such that P(a)=0 then. (z-a) is a solution.

 

[John Creighto]

no I meant the factor theorem... If you have a polynomial P(z) such that P(a)=0 then. (z-a) is a solution.

Yeah, but that only tells you how to check if something is a root. It doesn't tell you what to guess.

 

[||spoon||]

sorry i misread your original thread, my bad. Very sorry.