An elliptic fourth-order differential operator Aε on L2(Rd;Cn) is studied. Here ε>0 is a small parameter. It is assumed that the operator is given in the factorized form Aε=b(D)∗g(x/ε)b(D), where g(x) is a Hermitian matrix-valued function periodic with respect to some lattice and b(D) is a matrix second-order differential operator. We make assumptions ensuring that the operator Aε is strongly elliptic. The following approximation for the resolvent (Aε+I)−1 in the operator norm of L2(Rd;Cn) is obtained: (Aε+I)−1=(A0+I)−1+εK1+ε2K2(ε)+O(ε3). Here A0 is the effective operator with constant coefficients and K1 and K2(ε) are certain correctors.