I wrote something like below, it seems to do what I want - though not sure
if it's correct or not.
def simplify_common_factors_(p):
common_factors = [f**occurences for (f,occurences) in p.factor_list()
if f.number_of_operands()==0]
rs = p.divide_both_sides(prod(common_factors))
rs = rs.expand()
return rs
sage: simplify_common_factors_(3*x^2*y*z + x*y*z + y*z == 0)
3*x^2 + x + 1 == 0
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