The relation between GCD/LCM of 3 polynomials and the factors decomposition?

Question

Is it true that : the GCD of 3 polynomials is the product of the common factors (after factorization) of the these 3 polynomials ?

and the LCM of 3 polynomials is the product of all the common factors of the 3 polynomials ?

Answer 1

In a polynomial ring over a field as well as in $\mathbf Z$ (actually in any U.F.D.):

• The g.c.d. of two (hence any number of) polynomials is the product of the irreducible factors of each of them, with the lowest exponent. As consequence, only the common irreducible do appear in the g.c.d.
• The l.c.m. of two (hence any number of) polynomials is the product of the irreducible factors of each of them, with the highest exponent.

Example, as suggested by @amWhy:

$\gcd(x^2-1, x+1, x^2+2x+1)=\gcd\bigl((x-1)(x+1), x+1, (x+1)^2\bigr)=x+1$.

$\operatorname{lcm}(x^2-1, x+1, x^2+2x+1)=(x-1)(x+1)^2$.