Factoring Polynomials: Start Here

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    Factoring Polynomial
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Discussion Overview

The discussion focuses on the process of factoring the polynomial expression $m^8-n^8-2m^6n^2+2n^6m^2$. Participants explore various methods and approaches to factor the polynomial, engaging in a collaborative problem-solving effort.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant asks how to start factoring the polynomial.
  • Another participant suggests factoring the first two terms as a difference of squares and factoring out a common factor from the last two terms.
  • A participant presents their progress in factoring, showing intermediate steps and results.
  • Another participant proposes a different path for factoring, indicating that one of the factors may be a square of a binomial.
  • Two participants agree on a final factored form of the polynomial, stating it as $(m-n)^{3}(m+n)^{3}(m^2+n^2)$.

Areas of Agreement / Disagreement

There is a general agreement on the final factored form of the polynomial among some participants, while earlier steps and methods show varying approaches without explicit consensus on the best method.

Contextual Notes

Participants' contributions include various intermediate steps and methods, but the discussion does not resolve the optimal approach to factoring the polynomial or clarify all assumptions made during the process.

paulmdrdo1
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how would i start factoring this

$m^8-n^8-2m^6n^2+2n^6m^2$
 
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Re: factoring polynomial

You really need to post your attempts at these, this way we can see where you are at. :D

I would look at factoring the first two terms as the difference of squares, and the last two terms have a common factor as well (I would factor out $-2m^2n^2$)...what do you get?
 
Re: factoring polynomial

this is where i can get to

$(m^4+n^4)(m^4-n^4)-2n^2m^2(m^4-n^4)$

$(m^2-n^2)(m^2+n^2)(m^4+n^4)-2n^2m^2(m^2-n^2)(m^2+n^2)$

$(m-n)(m+n)(m^2+n^2)(m^4+n^4)-2n^2m^2(m-n)(m+n)(m^2+n^2)$
 
Re: factoring polynomial

I would take this path:

$$\left(m^4+n^4 \right)\left(m^4-n^4 \right)-2m^2n^2\left(m^4-n^4 \right)$$

$$\left(m^4-n^4 \right)\left(m^4-2m^2n^2+n^4 \right)$$

Do you recognize that the second factor is a square of a binomial?
 
Re: factoring polynomial

yes this is answer

$(m-n)^{3}(m+n)^{3}(m^2+n^2)$
 
Re: factoring polynomial

paulmdrdo said:
yes this is answer

$(m-n)^{3}(m+n)^{3}(m^2+n^2)$

Yes, good job! (Sun)
 

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