For an integer n ≥ 2 , let p ( z ) = ∏ k = 1 n ( z − α k ) and q ( z ) = ∏ k = 1 n ( z − β k ) , where α k , β k are real. We find the number of connected components of the real algebraic curve { ( x , y ) ∈ ℝ 2 : | p ( x + i y ) | − | q ( x + i y ) | = 0 } for some α k and β k . Moreover, in these cases, we show that each connected component contains zeros of p ( z ) + q ( z ) , and we investigate the locus of zeros of p ( z ) + q ( z ) .