Quartic polynomials

  • Thread starter chwala
  • Start date
  • #1
chwala
Gold Member
1,994
258

Homework Statement


Given ## x^4+x^3+Ax^2+4x-2=0## and giben that the roots are ## 1/Φ, 1/Ψ, 1/ξ ,1/φ##
find A


Homework Equations




The Attempt at a Solution


## (x-a)(x-b)(x-c)(x-d)=0## where a,b,c and d are the roots
 
Last edited by a moderator:

Answers and Replies

  • #2
mathwonk
Science Advisor
Homework Helper
11,391
1,626
whats your question?
 
  • #3
36,448
8,418

Homework Statement


Given ## x^4+x^3+Ax^2+4x-2=0## and giben that the roots are ## 1/Φ, 1/Ψ, 1/ξ ,1/φ##
find A


Homework Equations




The Attempt at a Solution


## (x-a)(x-b)(x-c)(x-d)=0## where a,b,c and d are the roots
Rather than work with all those Greek letters, I would make these substitutions:
Let ##a = \frac 1 \Theta, b = \frac 1 \Psi, c = \frac 1 \xi,d = \frac 1 \phi##.
Next, multiply out your equation. Then equate the coefficients of the ##x^3, x^2, x## terms and the constant term with those given in the original equation. Doing this, you should get four equations in the unknowns a, b, c, and d.

For example, one of the equations is ##(-a)(-b)(-c)(-d) = -2##, or equivalently, ##abcd = -2##.
 
  • #4
chwala
Gold Member
1,994
258
that's what i did...let me post my equations,
##abc +abd+acd+bcd = 0.5##
## ab+ad+ac+bd+bc+cd=-0.5A##
## a +b+c+d = 2##
 
  • #5
36,448
8,418
that's what i did...let me post my equations,
##abc +abd+acd+bcd = 0.5##
## ab+ad+ac+bd+bc+cd=-0.5A##
## a +b+c+d = 2##
It looks like you're on the right track, but I haven't worked the problem, so can't confirm that your equations are correct. With those three equations and the one from me, you have four equations in four unknowns, so with some work a solution can be found.
 
  • #6
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,722
that's what i did...let me post my equations,
##abc +abd+acd+bcd = 0.5##
## ab+ad+ac+bd+bc+cd=-0.5A##
## a +b+c+d = 2##

Where do all the "0.5"s come from?
 
  • #7
chwala
Gold Member
1,994
258
Ray just from a summary of my working. Anyway without boring you guys i realize that somewhere in the working one has to make use of the identity
## (a+b+c+d)^2 ≡ a^2 + b^2 +c^2 +d^2 + 2(ac+ad+bc+bd+cd+ab)## without which you can't arrive at the solution. Are there alternative methods?
## A=-1##
 
  • #8
chwala
Gold Member
1,994
258
Any other alternative method to the quartic polynomial?
 
  • #9
FactChecker
Science Advisor
Gold Member
7,441
3,219
you have four equations in four unknowns, so with some work a solution can be found.
I'm not so sure. These are not linear equations.
 
  • #10
36,448
8,418
you have four equations in four unknowns, so with some work a solution can be found.
I'm not so sure. These are not linear equations.
I don't see how that makes a difference other than you can't use matrix methods to find a solution.
Here's a simple example of two nonlinear equations:
##x^2 + y^2 = 1##
##(x - 1)^2 + y^2 = 1##
These equations represent two circles of radius 1. The first is centered at the origin, and the second is centered at (1, 0). By subtracting the first equation from the second, you get ##2x = 1## or ##x = \frac 1 2##. Back-substitution into the first equation yields ##y = \pm \frac {\sqrt 3} 2##, making the intersection points ##(\frac 1 2, \frac {\sqrt 3} 2)## and ##(\frac 1 2, \frac {-\sqrt 3} 2)##.
 
  • #11
FactChecker
Science Advisor
Gold Member
7,441
3,219
I don't see how that makes a difference other than you can't use matrix methods to find a solution.
Here's a simple example of two nonlinear equations:
##x^2 + y^2 = 1##
##(x - 1)^2 + y^2 = 1##
These equations represent two circles of radius 1. The first is centered at the origin, and the second is centered at (1, 0). By subtracting the first equation from the second, you get ##2x = 1## or ##x = \frac 1 2##. Back-substitution into the first equation yields ##y = \pm \frac {\sqrt 3} 2##, making the intersection points ##(\frac 1 2, \frac {\sqrt 3} 2)## and ##(\frac 1 2, \frac {-\sqrt 3} 2)##.
Ok. But I just don't think that there is a reliable theory regarding the existence of solutions to a number of nonlinear simultaneous equations. That being said, there might be something about these equations that can be used. I don't know.
 
  • #12
kuruman
Science Advisor
Homework Helper
Insights Author
Gold Member
2021 Award
12,311
5,405

Homework Statement


Given ## x^4+x^3+Ax^2+4x-2=0## and giben that the roots are ## 1/Φ, 1/Ψ, 1/ξ ,1/φ##
find A
I must be having a blind spot with this problem and it's driving me nuts. If the roots are given and they are truly roots, then isn't it true that substituting anyone of them for ##x## will give an equation that can be easily solved to get a consistent value for A? What am I missing that the problem is asking?
 
  • Like
Likes Delta2, chwala and FactChecker
  • #13
FactChecker
Science Advisor
Gold Member
7,441
3,219
I must be having a blind spot with this problem and it's driving me nuts. If the roots are given and they are truly roots, then isn't it true that substituting anyone of them for ##x## will give an equation that can be easily solved to get a consistent value for A? What am I missing that the problem is asking?
Ha! Of course! I am the blind one.
 
  • #14
chwala
Gold Member
1,994
258
I must be having a blind spot with this problem and it's driving me nuts. If the roots are given and they are truly roots, then isn't it true that substituting anyone of them for ##x## will give an equation that can be easily solved to get a consistent value for A? What am I missing that the problem is asking?
maybe, you could post your attempt, and see where you're not getting it...
 
  • #15
chwala
Gold Member
1,994
258
I must be having a blind spot with this problem and it's driving me nuts. If the roots are given and they are truly roots, then isn't it true that substituting anyone of them for ##x## will give an equation that can be easily solved to get a consistent value for A? What am I missing that the problem is asking?
did you manage to find the solution or you would like me to post it for you?
 
  • #16
kuruman
Science Advisor
Homework Helper
Insights Author
Gold Member
2021 Award
12,311
5,405
did you manage to find the solution or you would like me to post it for you?
It is against forum rules for me to post the solution. You are the OP, therefore you should post the solution and mark the problem as "solved". If my solution disagrees with yours, I will say so.
 
  • #17
chwala
Gold Member
1,994
258
I had already solved this problem, going through the threads, you seem not to understand, but you've confirmed that you know the solution. I don't think the question is still pending?
 
  • #18
kuruman
Science Advisor
Homework Helper
Insights Author
Gold Member
2021 Award
12,311
5,405
In post #4 you say
##ab+ad+ac+bd+bc+cd=-0.5A##
Is your solution then
##A=-2(ab+ad+ac+bd+bc+cd)=-2(\frac{1}{\Theta \Psi}+\frac{1}{\Theta \phi}+\frac{1}{\Theta \xi}+\frac{1}{\Psi \phi}+\frac{1}{\Psi \xi}+\frac{1}{\xi \phi})##?
The question is still pending until you are satisfied that you have the correct solution.
 
  • #19
chwala
Gold Member
1,994
258
I solved this in (post 7). I did not want to write the whole workings, let me check my files for this. It is solved already by me.
 
  • #20
kuruman
Science Advisor
Homework Helper
Insights Author
Gold Member
2021 Award
12,311
5,405
I solved this in (post 7).
In post #7 you say ##A=-1##. How can you get a numerical value for ##A## if you do not have numerical values for the roots ##\Theta##, ##\Psi##, ##\phi## and ##\xi##?
I did not want to write the whole workings, let me check my files for this. It is solved already by me.
Please check your files and post your solution in the form you think is right.
 
  • #21
chwala
Gold Member
1,994
258
ok, i am getting this equations, i can't see my files...i let the roots to be ;##a,b,c, d##
##(x-a)(x-b)(x-c)(x-d)= x^4+x^3+Ax^2+4x-2##
##a+b+c+d=-1##..........1
##ab+ac+ad+bc+bd+cd=A##......2
##bcd+acd+abd+abc= -4##.......3
##abcd= -2##...........4
is this step correct?
and further,
##(a+b+c+d)^2=a^2+b^2+c^2+d^2+2A##
 
  • #22
kuruman
Science Advisor
Homework Helper
Insights Author
Gold Member
2021 Award
12,311
5,405
That is correct. How much of all this do you need to get ##A##?
 
  • #23
chwala
Gold Member
1,994
258
am getting;
##(abcd)^2=4##.............5
## {a^2+b^2+c^2+d^2}=1-2A##..........6
i need way forward...
 
  • #24
kuruman
Science Advisor
Homework Helper
Insights Author
Gold Member
2021 Award
12,311
5,405
am getting;
##(abcd)^2=4##.............5
## {a^2+b^2+c^2+d^2}=1-2A##..........6
i need way forward...
So you have 4 equations in post #21 and 2 more in post #23 for a total of 6. Your goal is to find an expression for ##A##, meaning that you need to have ##A## alone on the left hand side and some expression on the right hand side involving ##a##, ##b##, ##c## and ##d##. Study each equation carefully, one at a time. How do you think you should proceed to achieve your goal?
 
  • #25
chwala
Gold Member
1,994
258
this is a tough one...still struggling, my latest attempt
## -(1/a +1/b+1/c+1/d)= abcd##...attempt 1
and
##-(cd(a+b)+ab(c+d)=4##
##cd(-1-c-d)##+##\frac 2 {cd} (c+d)=4##...attempt 2, am i on the right path?
 
  • #26
kuruman
Science Advisor
Homework Helper
Insights Author
Gold Member
2021 Award
12,311
5,405
this is a tough one...still struggling, my latest attempt
## -(1/a +1/b+1/c+1/d)= abcd##...attempt 1
and
##-(cd(a+b)+ab(c+d)=4##
##cd(-1-c-d)##+##\frac 2 {cd} (c+d)=4##...attempt 2, am i on the right path?
It is not as tough as you think. Take a deep breath, clear your mind then read very carefully the following that I repeat from post #24.
Your goal is to find an expression for ##A##, meaning that you need to have ##A## alone on the left hand side and some expression on the right hand side involving ##a##, ##b##, ##c## and ##d##.
Can you achieve this goal by looking at equations that do not contain ##A##?
 
  • #27
chwala
Gold Member
1,994
258
Thanks for your insight, let me look at it again.
 
  • #28
chwala
Gold Member
1,994
258
Lol still getting stuck...i will post my attempts...came up with equation...
## \frac {-2} {cd} ####(c+d)+cd(a+b)##= -4......7

another attempt: simultaneous equations,
## 1/a+1/b+1/c+1/d =2##............8
## a+b+c+d = -1##

another attempt:simultaneous equation,
## 1/a^2+1/b^2+1/c^2+1/d^2 + 2/A = 4##.......9
## a^2+b^2+c^2+d^2+2A = 1##
kindly advise if i am on the right track.
 
  • #29
kuruman
Science Advisor
Homework Helper
Insights Author
Gold Member
2021 Award
12,311
5,405
It seems we are talking past each other. Please explain to me, in your own words, what the problem is asking you to find. Once we agree on that, I will guide you to the next step.
 
  • #30
chwala
Gold Member
1,994
258
The problem requires that we find a numeric value for ##A## or rather the value of ##A## which is numeric ...
 
  • #31
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,722
The problem requires that we find a numeric value for ##A## or rather the value of ##A## which is numeric ...

Yes, but don't forget you are pretending that you know the values of the four roots, so you need a formula that, somehow, involves those roots.
 
  • #32
kuruman
Science Advisor
Homework Helper
Insights Author
Gold Member
2021 Award
12,311
5,405
To illustrate what @Ray Vickson said, suppose I told you that
##\Theta = -5.50427##
##\Psi=0.08167 + 0.27755 i##
##\phi=0.08167 - 0.27755 i##
##\xi=4.34094##
Can you find a numeric value for ##A## given this set of roots? Note that there are infinitely many sets of roots because ##A## can be chosen to have infinitely many values.

On edit (2 days later)
Actually, the numbers above are the actual roots consistent with the problem's statement. In other words
##1/\Theta = -5.50427##
##1/\Psi=0.08167 + 0.27755 i##
##1/\phi=0.08167 - 0.27755 i##
##1/\xi=4.34094##

I apologize for the confusion.
 
Last edited:
  • #33
chwala
Gold Member
1,994
258
let me look at it again...
 
  • #34
chwala
Gold Member
1,994
258
so reading your comments, and from my understanding, the only possibility is to use trial and error in trying to figure out the roots of the problem, this is my latest equation.
## \frac {-2} {cd}####(c+d) + cd(-1-c-d)=-4##
is the above equation correct? if so, then how do we get the values of ##c## and ##d## ?
 
  • #35
kuruman
Science Advisor
Homework Helper
Insights Author
Gold Member
2021 Award
12,311
5,405
so reading your comments, and from my understanding, the only possibility is to use trial and error in trying to figure out the roots of the problem, this is my latest equation.
## \frac {-2} {cd}####(c+d) + cd(-1-c-d)=-4##
is the above equation correct? if so, then how do we get the values of ##c## and ##d## ?
What roots? I gave you the roots for one choice of ##A## in #32. Let me remind you of the definitions
Rather than work with all those Greek letters, I would make these substitutions:
Let ##a = \frac 1 \Theta, b = \frac 1 \Psi, c = \frac 1 \xi,d = \frac 1 \phi##.
So you know ## \Theta##, ##\Psi##, ##\xi## and ##\phi## and you can easily find ## a##, ##b##, ##c## and ##d## from the definitions.
Can you find ##A##?
 

Suggested for: Quartic polynomials

Replies
15
Views
720
  • Last Post
Replies
5
Views
340
  • Last Post
Replies
12
Views
1K
Replies
9
Views
698
Replies
11
Views
2K
Replies
5
Views
626
Replies
18
Views
686
  • Last Post
Replies
1
Views
720
Replies
7
Views
758
Top