A one-dimensional model of the coherent amplification in a high- and swept-gain amplifier in the presence of strong spontaneous emission is presented and applied to seeding an x-ray laser with a high harmonic pulse. The atomic structure of the amplifying medium includes level degeneracy, and the Maxwell-Bloch equations are used to describe the propagation and interaction processes. This analysis elucidates, in the most comprehensive way, the influence of the random spontaneous emission on the coherent amplification process, as reflected in the basic pulse parameters such as its shape, polarization, and coherence.