GCD of Polynomials: Is 1 Always the Solution?

tgt · Nov 21, 2008

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

tiny-tim · Nov 21, 2008

tgt said:

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

Hi tgt!

Hint: if the roots of the first one are p and q, what are the roots of the second one?

tgt · Nov 21, 2008

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.

tiny-tim · Nov 22, 2008

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?

tgt · Nov 22, 2008

tiny-tim said:

What about x² + 3x + 2 = 0 and (x + 1)² + 3(x + 1) + 2 = 0?

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

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