First note that ''f'' and ''g,'' and therefore also ''h, k'' and all the other polynomials in the ideal ''I'' have the following three zeroes in the (''x'',''y'') plane in common, as indicated in the figure: {(1,1),(-1,1),(0,0)}.
with ''c'' a nonzero rational number and ''p'' a polynomial in the variable ''y'' only; the reason being that such an ''m'' can never have two distinct zeroes with the same value for ''y'' (in this case, the points (1,1) and (-1,1)).