In this paper, a synthesis method by applying the Taylor series is proposed to design a variety of low-pass mixed lumped and distributed circuit networks, including the low-pass filters and low-passing impedance transformers. First, the principle of applying the Taylor polynomials for these mixed circuit networks is discussed. Herein, the trigonometric functions, commonly used to characterize distributed circuits, are expressed in the form of polynomials by the application of Taylor polynomials. Therefore, the transfer function of the mixed networks is simplified from a multivariable to a polynomial form. Following this, a generic circuit structure is proposed to design either a low-pass filter or a low-passing impedance transformer with specified port impedances. The desired filtering function uses the similar method to transform the trigonometric function into a format of polynomials. Under the equalization of the transfer function and filtering function, all the unknown design parameters are obtained. As two examples, a low-pass filter and a low-passing impedance transformer are synthesized, designed, and fabricated to experimentally verify the proposed design approach.