# Quartic polynomials

Gold Member

## 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

## 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:

Homework Helper

Mentor

## 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

## 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##.

chwala
Gold Member
that's what i did...let me post my equations,
##abc +abd+acd+bcd = 0.5##
## a +b+c+d = 2##

Mentor
that's what i did...let me post my equations,
##abc +abd+acd+bcd = 0.5##
## 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.

chwala
Homework Helper
Dearly Missed
that's what i did...let me post my equations,
##abc +abd+acd+bcd = 0.5##
## a +b+c+d = 2##

Where do all the "0.5"s come from?

chwala
Gold Member
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##

Gold Member
Any other alternative method to the quartic polynomial?

Gold Member
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.

Mentor
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)##.

scottdave
Gold Member
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.

Homework Helper
Gold Member
2021 Award

## 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?

Delta2, chwala and FactChecker
Gold Member
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.

chwala
Gold Member
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...

Gold Member
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?

Homework Helper
Gold Member
2021 Award
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.

Gold Member
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?

Homework Helper
Gold Member
2021 Award
In post #4 you say
##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.

Gold Member
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.

Homework Helper
Gold Member
2021 Award
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.

Delta2
Gold Member
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
##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##

Homework Helper
Gold Member
2021 Award
That is correct. How much of all this do you need to get ##A##?

Gold Member
am getting;
##(abcd)^2=4##.............5
## {a^2+b^2+c^2+d^2}=1-2A##..........6
i need way forward...

Homework Helper
Gold Member
2021 Award
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?

Gold Member
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?

Homework Helper
Gold Member
2021 Award
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##?

Gold Member
Thanks for your insight, let me look at it again.

Gold Member
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.

Homework Helper
Gold Member
2021 Award
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.

Gold Member
The problem requires that we find a numeric value for ##A## or rather the value of ##A## which is numeric ...

Homework Helper
Dearly Missed
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.

Homework Helper
Gold Member
2021 Award
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:
Gold Member
let me look at it again...

Gold Member
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## ?

Homework Helper
Gold Member
2021 Award
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##?