GCD of Polynomials: Is 1 Always the Solution?

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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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