A new method is presented for constructing the irreducible representation and for calculating the Clebsch-Gordan coefficient of the quantum algebra su_q(3). The method is different from others and does not depend on any concrete realization of su_q(3). This means that the result holds in general. A set of recurrence formula has been obtained for calculating the Clebsch-Gordan coefficient. In fact the recurrence formula is aimed at the so-called scalar factor of su_q(3), and means that the Racah factorization lemma also holds for the quantum algebra su_q(3).