Polynomial equation with complex answers

Click For Summary

Homework Help Overview

The discussion revolves around a polynomial equation of degree four, specifically x4 + 4x3 - 7x2 - 12x + 24 = 0. The original poster seeks to understand how to determine that the solutions are complex without relying on calculators or external tools.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants explore various methods such as completing the square and using the Rational Root Theorem. Questions arise about the definition of "complex solutions" and the process to identify them without computational aids.

Discussion Status

There is an active exploration of the problem with participants suggesting different approaches, including differentiation to find points of inflection and analyzing the behavior of the polynomial's graph. Some participants express uncertainty about the nature of the solutions, while others provide insights into the characteristics of the polynomial.

Contextual Notes

Participants mention the use of online calculators as a potential shortcut, which raises questions about the appropriateness of such tools in the context of the discussion. The Rational Root Theorem is referenced as a means to demonstrate the absence of rational solutions.

axon23
Messages
4
Reaction score
0
x4+4x3-7x2-12x+24=0


Ive been working on this with my brother for a long time and we don't need to find an answer. we just need to reach a point when we can say that the solution is complex. Any ideas? Any nudge in the right direction with be great.


We tried completing squares, bruteforcing the answer and many other failed attempts
 
Physics news on Phys.org
What do you mean by "the solution is complex"? There are 4 roots to this. If you find the roots using some online calculators (essentially cheating), you'll see that 2 of them are real and other 2 are complex.
 
yea but what if i wanted to figure out the answer without a calculator. how would the process be done?
 
points of inflection

axon23 said:
x4+4x3-7x2-12x+24=0

Hi axon23! :smile:

Differentiate twice to find the points of inflection …

then you should be able to sketch the graph. :smile:
 
I'm not sure what you mean by complex solution but it can be shown that there are no rational solutions by the Rational Root Theorem.
 
Set the derivative equal to 0 to find max and min points. Since this even power, if all minimum values are positive, the graph does not cross the x-axis and there are no real roots.
 

Similar threads

Showing a polynomial is not solvable by radicals
  • · Replies 18 ·
Replies
18
Views
5K
Undergrad Quintic Polynomial Tschirnhaus Transform: Possible complex Bring Radicals
  • · Replies 0 ·
Replies
0
Views
3K
Showing that one polynomial divides another
  • · Replies 8 ·
Replies
8
Views
2K
Hey where do i begin with this complex polynomial question?
  • · Replies 12 ·
Replies
12
Views
2K
Factoring Polynomials [Abstract Algebra]
  • · Replies 17 ·
Replies
17
Views
2K
Solution to real and complex polynomial function?
  • · Replies 1 ·
Replies
1
Views
2K
Show that complex conjugate is also a root of polynomial with real coefficients
  • · Replies 3 ·
Replies
3
Views
7K
How Can Phasors Represent the Function g(t) in Complex Form?
  • · Replies 31 ·
2
Replies
31
Views
3K
How Do You Form a Degree 4 Polynomial with Given Complex Zeros?
  • · Replies 7 ·
Replies
7
Views
2K
Writing a polynomial in terms of other polynomials (Hermite, Legendre, Laguerre)
  • · Replies 8 ·
Replies
8
Views
9K