Complex Number equations from roots

Click For Summary

Discussion Overview

The discussion revolves around determining the real values of coefficients a, b, c, and d in the polynomial equation z4 + az3 + bz2 + cz + d = 0, given that the complex numbers z1 = 3 + j and z2 = -5 + 5j are roots. The scope includes homework-related problem-solving and mathematical reasoning.

Discussion Character

  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant expresses uncertainty about how to start solving the problem, mentioning familiarity with converting between Cartesian coordinates and Euler's formula.
  • Another participant hints that if z1 is a root of a real-coefficient equation, there should be a corresponding conjugate root.
  • A participant acknowledges the hint and indicates they now understand the need for the conjugate root.
  • One participant questions how to assign calculated values to the coefficients a, b, c, and d after multiplying the roots and their conjugates, noting a specific value of 500 obtained from this multiplication.
  • Another participant confirms that the value of 500 would correspond to d and suggests a more efficient method to find the polynomial by multiplying the factors associated with the roots directly.
  • A participant expresses gratitude for the guidance and reflects on the importance of treating z as a variable rather than just seeking numerical values.

Areas of Agreement / Disagreement

Participants generally agree on the approach of using conjugate roots for real-coefficient polynomials and the method of multiplying factors to find the polynomial. However, there is no consensus on the specific assignment of values to the coefficients a, b, c, and d, as this remains a point of inquiry.

Contextual Notes

There are limitations regarding the assumptions about the roots and their conjugates, as well as the dependence on the properties of polynomials with real coefficients. The discussion does not resolve how to systematically derive the coefficients from the roots.

DmytriE
Messages
78
Reaction score
0

Homework Statement


Determine the only real values a, b, c, and d such that the equation:
z4+az3+bz2+cz+d = 0​
has both z1 and z2 as roots.

z1 = 3 + j
z2 = -5 + 5j

Homework Equations


z = x + yj.
z = |z|ej\theta


The Attempt at a Solution


I am not sure where to begin. I can convert between Cartesian coordinates and Euler's formula but don't know where to go from there.

Any help would be greatly appreciated!
 
Physics news on Phys.org
Hi DmytriE! :smile:

Hint: if 3 + j is a root (of a real-coefficient equation), can you spot another root? :wink:
 
tiny-tim said:
can you spot another root? :wink:

Indeed! Now, I do. Thanks for the hint!
 
Last edited:
I have calculated the different combination of roots but how do I know which variable (a,b,c,d) the values go to? When I multiplied z1, z1*, z2, and z2* I got a value of 500. Would that go into d?

It seems logical when I compare it to a second degree polynomial...
 
DmytriE said:
When I multiplied z1, z1*, z2, and z2* I got a value of 500. Would that go into d?

Yes.

But why not do it all in one go? …

Find (z - z1)(z - z1*), then (z - z2)(z - z2*), then multiply them. :wink:
 
tiny-tim said:
Yes.

But why not do it all in one go? …

Find (z - z1)(z - z1*), then (z - z2)(z - z2*), then multiply them. :wink:

Great! Thank you again. I keep forgetting that z is a variable so I was strictly looking for the number rather than the equation with the coefficients.
 

Similar threads

Complex numbers rectangular form
  • · Replies 11 ·
Replies
11
Views
13K
Engineering AC Circuit Phasors: Find I1, I2, I3
  • · Replies 26 ·
Replies
26
Views
3K
When to transform the current source to voltage source?
  • · Replies 1 ·
Replies
1
Views
2K
How Do You Solve Complex Number Equations in Quadratics and Exponentials?
  • · Replies 39 ·
2
Replies
39
Views
6K
High School Principal square root of a complex number, why is it unique?
  • · Replies 13 ·
Replies
13
Views
6K
Engineering Deriving the Cable Equation (neuroscience) from Fundamental Physics Laws
  • · Replies 1 ·
Replies
1
Views
4K
Do These Functions Qualify as Group Homomorphisms?
  • · Replies 3 ·
Replies
3
Views
2K
Is (y + z)x1 = (y1 + z1)x the equation of a straight line in 3D space?
  • · Replies 17 ·
Replies
17
Views
3K
How to get roots of this complex equation
  • · Replies 9 ·
Replies
9
Views
3K
Complex constant from single root.
  • · Replies 1 ·
Replies
1
Views
2K