Can anyone me to generalize and prove this if it is valid?

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Discussion Overview

The discussion revolves around a mathematical expression involving polynomial identities and their generalization for odd prime values of n. Participants explore the validity of the proposed equations and seek assistance in proving them, with a focus on binomial expansions and formatting mathematical expressions correctly.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Homework-related, Mathematical reasoning

Main Points Raised

  • One participant presents a polynomial identity and requests proof for a general case involving odd primes.
  • Another participant suggests that the original post is difficult to read due to formatting issues and advises using LaTeX or attaching a PDF for better clarity.
  • A different participant recalls a similar post from the recent past and recommends learning about the binomial theorem to aid in expanding the expressions.
  • Another participant proposes that the problem is a simple rearrangement of binomial expansions and encourages using the general form of binomial expansions for proof.
  • One participant offers a formatting tip using "sup" and "sub" tags for creating superscripts and subscripts in the forum.
  • A later reply acknowledges the feedback received from other participants.

Areas of Agreement / Disagreement

There is no clear consensus on the validity of the proposed mathematical identities or the best approach to prove them. Multiple suggestions and perspectives are presented without resolution.

Contextual Notes

Some participants express difficulty in reading the mathematical expressions due to formatting, which may affect the clarity of the discussion. The reliance on binomial expansions is noted, but specific steps or assumptions in the proof remain unresolved.

mahmudarif
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X^4-(X^3)y+(X^2)(y^2)-x(y^3)+y^4= (x+y)^4-5xy(x+y)^2+5(xy)^2

x^6-(x^5)y+(X^4)(y^2)-(X^3)(y^3)+(X^2)(y^4)-x(y^5)+y^6=(x+y)^6-7xy(x+y)^4 +14((xy)^2)(x+y)^2 - 7(xy)^3

......
If n is an odd prime then prove,

x^n-1 - X^(n-2).y+...-x.y^(n-2)+y^(n-1) = (x+y)^n-1 - nxy(x+y)^(n-3) +...(-1)^((n-1)/2) . n .(xy) ^ ((n-1)/2)


Thank you very much in advance for your assistance.
 
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mahmudarif said:
X^4-(X^3)y+(X^2)(y^2)-x(y^3)+y^4= (x+y)^4-5xy(x+y)^2+5(xy)^2

x^6-(x^5)y+(X^4)(y^2)-(X^3)(y^3)+(X^2)(y^4)-x(y^5)+y^6=(x+y)^6-7xy(x+y)^4 +14((xy)^2)(x+y)^2 - 7(xy)^3

......
If n is an odd prime then prove,

x^n-1 - X^(n-2).y+...-x.y^(n-2)+y^(n-1) = (x+y)^n-1 - nxy(x+y)^(n-3) +...(-1)^((n-1)/2) . n .(xy) ^ ((n-1)/2)


Thank you very much in advance for your assistance.



Your post is very difficult to read in ASCII...I even didn't try. My advice: learn how to post here in LaTeX or else

attach some document, preferably PDF, where mathematical stuff appears decently.

DonAntonio
 
I think this has been posted before recently but, for the love of me, I can't find the original post.

And I believe the suggestion was to learn about the binomial theorem; then try to expand (x+y)^4 and (x+y)^6.
 
that's simple re arrangement of binomial expansions. for general proof, try general form of binomial expansions. you will get it easily.
 
Or else use the "sup" and "sub" tags to create superscripts and subscripts:

[noparse]xy xy[/noparse]

xy xy
 
Got the point. Thanks every one...
 

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