Roots of a third degree polynomal equation (complex numbers)

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Homework Help Overview

The discussion revolves around finding the roots of a third-degree polynomial equation with complex coefficients, specifically focusing on the equation z^3 + az^2 + bz + c, where the roots form a triangle with a specified area in the complex plane. One known root is 1 + 3i, and the problem involves determining the other roots as well as the coefficients a, b, and c.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation, Assumption checking

Approaches and Questions Raised

  • Participants discuss the implications of having complex roots and the relationship between the roots and the area of the triangle they form. There is exploration of how to calculate the area based on the known roots and the geometric properties of triangles in the complex plane.

Discussion Status

Some participants have provided insights into the geometric interpretation of the problem, suggesting that the area of the triangle can be determined using the known roots. There is acknowledgment of the need to find the height of the triangle based on the base formed by the known roots. Multiple interpretations of the third root are being explored, with suggestions of possible values.

Contextual Notes

Participants note that the coefficients a, b, and c are real numbers, and there is a focus on the implications of this condition for the roots of the polynomial. The area of the triangle formed by the roots is a critical constraint in the discussion.

hrappur2
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Hey everyone! Here I have a problem I don't know how to solve so help would be greatly appreciated!

Homework Statement


Here is an equation z^3+az^2+bz+c where a, b and c are real numbers. If the roots are drawn in the complex plane they form a triangle with area of 9 units. One root of the equation is 1+3i.

(a) Find other roots of the equation.
(b) Find a, b and c2. The attempt at a solution
The only thing I know is that if 1+3i is a root than 1-3i is also a root but more I don't know, I have no idea how to solve this without a, b and c i don't know how to find them.

-Thanks in advance!

EDIT: I think I finally figured out how to solve this. I tried to insert (1+3i) and (1-3i) into the equation and got a=1, b=4 and c=30 then I put it in the original equation and get the third root, z=-3. Is that correct? What is confusing me is that the area has to be 9, I don't get that.
 
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hi hrappur2! :smile:
hrappur2 said:
EDIT: I think I finally figured out how to solve this. I tried to insert (1+3i) and (1-3i) into the equation and got a=1, b=4 and c=30 …

how? :confused:

it might be easier to start with the triangle …

you know the position of one of its sides …

so what line does the opposite point have to be on if the area is 9 ? :wink:​
 
Unfortunately, I can't see the equation itself but yes, if the equation has only real coefficients and 1+ 3i is a root then 1- 3i is another root and the third root must be real. The fact that it is real means it is at the point (x, 0) in the complex plane for some real number x. Can you determine the area of such a triangle, in terms of x?

Note tiny-tim's suggestion. The area of a triangle is "base times height" and we can take the line through 1+ 3i and 1- 3i ((1, 3) and (1, -3)) as base. What is its length? Knowing that and the fact that the area is 9, we can find the height of the triangle. The third vertex must lie on a line parallel to the line through (1, 3) and (1, -3) and a distance equal to the height of the triangle from it.
 
Thanks a lot for your help!

So if I'm understanding this correctly I draw 1-3i and 1+3i into the plane and get that the base is 6 so I have to have height 3 so the area is 9. So the third root must be either -2 or 4. Is that correct?
 
yes, the third root must be real, so it's -2 or 4 :smile:
 

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