Two different kinds of stability tests, or more generally zero location tests, are proposed when a real polynomial formulated in the delta-operator is given. The method used is a simple linear transformation based on some known discrete-time tests, including the Schur-Cohn test (SCT), and the proposed algorithms are computationally efficient, Furthermore, it is shown that as the sampling interval vanishes, all new algorithms converge to their continuous-time limits, therefore providing smooth transitions from the shift operator-based discrete-time tests to the continuous-time ones, One of these limits is the classic Routh test (RT), and the other is a new test, closely related to the RT. Examples are given to demonstrate the tests which show clear numerical advantages over the traditional shift-operator-based Ones such as the SCT in case of fast sampling. The significance of these new tests lies beyond the fact that they have provided efficient and reliable zero location tests directly in tile delta-domain. These tests provide the missing links between the discrete-time shift-operator-based tests and the continuous-time ones. As a consequence, two separate and yet related families of zero location tests are established, which include the well-known RT and the SCT.