In this paper, two factorizations for fourth-order tensors based on different multiplications of the fourth-order tensors are investigated. One is, called as F-TD, based on the fourth-order tensor multiplication (F-product). Another is, the fourth-order tensor multiplication and decomposition are defined, called as B-product and FT-SVD, based on the t-product of the third-order tensor multiplication.  Meanwhile, two tensor approximation theorems are present using two  decomposition methods. Finally, three numerical examples are given to demonstrate the accuracy and the feasibility of our proposed methods.