# Solving Polynomials with Complex Coefficients

• ||spoon||
In summary, the conversation discusses finding factors for polynomials with complex coefficients and whether there is a quicker method than using the factor theorem. There is also a question about the solutions of equations with complex coefficients and whether there is a rule for quickly solving them. The conversation ends with a mention of a possible solution using group theory or modern algebra.
||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||

Last edited:
||spoon|| said:
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.

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

||spoon|| said:
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.

## What are complex coefficients in a polynomial?

Complex coefficients in a polynomial refer to the numbers that are multiplied with the variable(s) in a polynomial expression. These numbers can be real or imaginary, or a combination of both.

## How do you solve a polynomial with complex coefficients?

To solve a polynomial with complex coefficients, you can use the same methods as solving a polynomial with real coefficients. This includes factoring, using the quadratic formula, or long division. However, you may need to use complex numbers and their properties in the process.

## What is the difference between real and complex solutions in a polynomial?

The difference between real and complex solutions in a polynomial is that real solutions are values that can be found on the real number line, while complex solutions are values that involve imaginary numbers and cannot be found on the real number line.

## Can a polynomial with complex coefficients have only real solutions?

Yes, a polynomial with complex coefficients can have only real solutions. This is because complex numbers can also be real numbers, so the solutions may not involve any imaginary numbers.

## Why is it important to consider complex coefficients when solving a polynomial?

It is important to consider complex coefficients when solving a polynomial because they can provide more accurate and precise solutions. Complex numbers allow us to solve equations that cannot be solved with only real numbers, and they have a wide range of applications in fields such as physics, engineering, and mathematics.

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