GCD of Polynomials: Is 1 Always the Solution?

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Discussion Overview

The discussion revolves around the greatest common divisor (GCD) of two specific polynomial expressions: x^2+x+c and (x-a)^2+(x-a)+c. Participants explore whether the GCD is always 1 and examine conditions under which polynomials may share roots.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions if the GCD of the two polynomials is always 1.
  • Another participant hints at the relationship between the roots of the two polynomials, suggesting that understanding the roots may clarify the GCD.
  • Some participants propose that two polynomials have a nonzero GCD if they share at least one root, implying that certain forms of polynomials will have a GCD of 1 due to the absence of shared roots.
  • A later reply challenges this by providing a specific example of two polynomials that appear to share roots, questioning the earlier assertion about GCD being 1.
  • One participant acknowledges a lapse in reasoning regarding the example provided, indicating uncertainty in their previous claim.

Areas of Agreement / Disagreement

Participants do not reach a consensus; multiple competing views remain regarding the conditions under which the GCD of the polynomials is 1.

Contextual Notes

Some assumptions about the nature of the roots and the specific forms of the polynomials may be missing or unclear, which could affect the conclusions drawn about the GCD.

tgt
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Would the GCD of x^2+x+c and (x-a)^2+(x-a)+c always be 1?
 
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tgt said:
Would the GCD of x^2+x+c and (x-a)^2+(x-a)+c always be 1?

Hi tgt! :smile:

Hint: if the roots of the first one are p and q, what are the roots of the second one? :wink:
 
So it seems that two polynomials have nonzero GCD when there is at leaste one root shared between the two. So any two polynomials of the form a(x-t)^2+b(x-t)+c and ax^2+bx+c must have GCD 1 since they wouldn't have any roots shared between them.
 
tgt said:
So it seems that two polynomials have nonzero GCD when there is at leaste one root shared between the two. So any two polynomials of the form a(x-t)^2+b(x-t)+c and ax^2+bx+c must have GCD 1 since they wouldn't have any roots shared between them.

What about x2 + 3x + 2 = 0 and (x + 1)2 + 3(x + 1) + 2 = 0? :rolleyes:
 
tiny-tim said:
What about x2 + 3x + 2 = 0 and (x + 1)2 + 3(x + 1) + 2 = 0? :rolleyes:

ok, I wasn't thinkng clearly at the time.
 

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