Finding Solutions for Polynomial Division: Where to Begin?

In summary: I'm not sure what you mean by "how to check the solutions". If you mean "how to check the values of p and q that we have found", you simply substitute them into the original polynomial division and verify that the division comes out with no remainder. If you mean "how to check that we have found all possible values of p and q", then that is an entirely different question that I am not able to answer.
  • #1
stungheld
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Homework Statement


How many pairs of solutions make x^4 + px^2 + q = 0 divisable by x^2 + px + q = 0

Homework Equations


x1 + x2 = -p
x1*x2= q[/B]

The Attempt at a Solution


I tried making z = x^2 and replacing but got nowhere. I figure 0,1,-1 are 3 numbers that fit but I am not sure what's being asked also. How many pairs of x1 and x2 make the polynomial 1 divisible by polynomial 2. How to start solving this?
 
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  • #2
The second polynomial can be written as [itex](x- a)(x- b)= x^2- (a+ b)x+ ab[/itex] with a+ b= -p and ab= q. It divides the first if and only if the first polynomial has those same factors: is of the form [itex](x- a)(x- b)(x- c)(x- d)= x^4- (a+ b+ c+ d)x^3+ (ab+ ac+ ad+ bc+ bd+ cd)x^2- (abc+ acd+ bcd)x+ abcd[/itex] with a+ b+ c+ d= 0, ab+ ac+ ad+ bc+ bd+ cd= p, abc+ acd+ bcd= 0, and abcd= q. See what you can make of those 6 equations.
 
  • #3
Just a guess:

Maybe the question is:
For how many pairs (p,q) is the polynomial ##x^4+px^2+q## divisible by the polynomial ##x^2+px+q##?
Where divisible means that the quotient is a polynomial too.

Admittedly, that doesn't explain the "=0" in the question, so I'm far for certain.

EDIT: didn't see the post above before I posted, so just ignore this.
 
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  • #4
stungheld said:

Homework Statement


How many pairs of solutions make x^4 + px^2 + q = 0 divisable by x^2 + px + q = 0

Homework Equations


x1 + x2 = -p
x1*x2= q[/B]

The Attempt at a Solution


I tried making z = x^2 and replacing but got nowhere. I figure 0,1,-1 are 3 numbers that fit but I am not sure what's being asked also. How many pairs of x1 and x2 make the polynomial 1 divisible by polynomial 2. How to start solving this?

This is a good question and will stretch your understanding of polynomials. First, I assume that you've added the "=0". It's important to know the difference between a polynomial and a polynomial equation. In this case you are dealing with polynomials. You are not trying to find the values of ##x## for which the polynomial is 0.

I would rephrase the question as:

Find ##p## and ##q## such that ##x^2 + px + q## divides ##x^4 + px^2 + q##

Can you see the first step?
 
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  • #5
PeroK said:
This is a good question and will stretch your understanding of polynomials. First, I assume that you've added the "=0". It's important to know the difference between a polynomial and a polynomial equation. In this case you are dealing with polynomials. You are not trying to find the values of ##x## for which the polynomial is 0.

I would rephrase the question as:

Find ##p## and ##q## such that ##x^2 + px + q## divides ##x^4 + px^2 + q##

Can you see the first step?

It is true what you said. I added = 0. It shouldn't be there. I don't seem to have any idea on how to start, maybe you could help me with some subtle hint without giving me the step?
 
  • #6
stungheld said:
It is true what you said. I added = 0. It shouldn't be there. I don't seem to have any idea on how to start, maybe you could help me with some subtle hint without giving me the step?

In general, the definition of ##A## divides ##B## is that there exists some ##C## such that ##B = AC##
 
  • #7
stungheld said:
I don't seem to have any idea on how to start, maybe you could help me with some subtle hint without giving me the step?
Do what you wrote in the title: Polynomial Long Division. Divide x4+px2+q with x2+px+q, and find the values of p and q so as the remainder is zero.
 
  • #8
I haven't done long division before so i checked it out online and tried to do it.Could you check this?
 

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  • #9
stungheld said:
I haven't done long division before so i checked it out online and tried to do it.Could you check this?
You have sign errors at the last stage.
 
  • #10
I don't see it
 
  • #11
stungheld said:
I haven't done long division before so i checked it out online and tried to do it.Could you check this?

(Your reply to ehild:
stungheld said:
I don't see it
)

upload_2015-11-21_12-18-8.png

Those are the last three lines of the image you posted.

You need to subtract the part enclosed in the red parentheses.
 
  • #12
(p^2 + p - q)(- px - q) + q(px + 1)= 0
This must be true for some p and q so that there is no remainder. But how do i find those values? There are three variables here.
 
  • #13
stungheld said:
(p^2 + p - q)(- px - q) + q(px + 1)= 0
This must be true for some p and q so that there is no remainder. But how do i find those values? There are three variables here.
They must be true for any value of x. Collect the x terms and the constant terms separately, in form A(p,q)x +B(p,q)≡0. Both A and B are equal to zero. So you have two equations with the parameters p,q.
 
  • #14
Well i get x(-p^3 - p^2 + 2pq) + (-qp^2 - qp + q^2 + q) = 0
So like you said A and B are 0. I set them to 0 and solved for p but got some cubic equation so i supposed three solutions there. But i can't solve for q.
 
  • #15
I suggest you avoid polynomial division altogether and use HallsofIvy's suggestion back in post #2. He made a slight error in the coefficient of the linear term, however. It should be (abc + abd + acd + bcd).

Consider the cases q = 0 and ##q \ne 0## separately.
 
  • #16
vela said:
I suggest you avoid polynomial division altogether and use HallsofIvy's suggestion back in post #2. He made a slight error in the coefficient of the linear term, however. It should be (abc + abd + acd + bcd).

Consider the cases q = 0 and ##q \ne 0## separately.
That seems more complicated. Any chance we could try resolving this polynomial division? How to check the solutions after the last part of the division? I separated the cases in the x(-p^3 - p^2 + 2pq) + (-qp^2 - qp + q^2 + q) = 0. I tried to eliminate one variable by substituting p for q but gotten -2p^3 + 2p^2 - p = 0, and when i try the other way around i get something like p^2 + p = q - 1
 
  • #17
stungheld said:
Well i get x(-p^3 - p^2 + 2pq) + (-qp^2 - qp + q^2 + q) = 0
So like you said A and B are 0. I set them to 0 and solved for p but got some cubic equation so i supposed three solutions there. But i can't solve for q.
You can factorize both equations, getting p(-p^2-p+2q)=0 and q(-p^2-p+q+1)=0. What solutions are possible?
 
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  • #18
ehild said:
You can factorize both equation, getting p(-p^2-p+2q)=0 and q(-p^2-p+q+1)=0. What solutions are possible?
p and q = 0
q = 0 and p either of two solutions
p = 0 and q = -1
p = q = 1
That makes it 5. Is that it?
 
  • #19
stungheld said:
p and q = 0
q = 0 and p either of two solutions
p = 0 and q = -1
p = q = 1
That makes it 5. Is that it?
Make a table for the possible values of p and q.
It looks that you counted p=0 q=0 twice, and one solution is still missing. What are the solutions if neither p nor q are zero?
 
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  • #20
How did i count p and q to be 0 twice? I counted both to be 0, two cases were in the first q is 0 and p p1 or p2 ( making the first expression 0) and the only case were p and q are non zero but still make zero are p = q = 1. I see no other way
 
  • #21
So you have so far
p=0, q=0
p=0, q=-1
p? , q=0
plus the solutions of
p^2+p-2q=0 ,
p^2+p+-q-1=0
one of them is p=1, q=1, what is the other one?
 
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  • #22
If q = 0 p = -1 and for the last one its p = -2 and q = 1
5 solutions would then be the answer
 
  • #23
Perfect!:smile:
 
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Related to Finding Solutions for Polynomial Division: Where to Begin?

What is polynomial division?

Polynomial division is a method used to divide polynomials, which are algebraic expressions that contain variables and coefficients. It involves dividing one polynomial by another polynomial.

What are the steps for polynomial division?

The steps for polynomial division are as follows:

  1. Arrange the polynomials in descending order of exponents.
  2. Divide the first term of the dividend by the first term of the divisor.
  3. Multiply the result by the divisor and write the answer below the dividend.
  4. Subtract the result from the corresponding terms of the dividend.
  5. Bring down the next term of the dividend.
  6. Repeat the process until all terms have been brought down and the remainder is smaller than the divisor.

What is the difference between polynomial division and long division?

The main difference between polynomial division and long division is that in polynomial division, both the dividend and divisor are polynomials, while in long division, the dividend can be any number or algebraic expression.

What is the purpose of polynomial division?

The purpose of polynomial division is to simplify complex expressions and to find the quotient and remainder when dividing one polynomial by another. It is also used in various mathematical and scientific calculations.

What are some common mistakes to avoid in polynomial division?

Some common mistakes to avoid in polynomial division include:

  • Not arranging the polynomials in descending order of exponents.
  • Making errors during multiplication or subtraction.
  • Forgetting to bring down the next term of the dividend.
  • Not simplifying the final answer.

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