In this paper, we discuss an algebraic decoding of BCH codes over symbol-pair read channels. The channels output overlapping pairs of symbols in storage applications. The pair distance and pair error are used in the channels. We define a polynomial that represents the positions of the pair errors as the error-locator polynomials and a polynomial that represents the positions of the pairs of a received pair vector in conflict as conflict-locator polynomial. In this paper, we propose algebraic methods for correcting two-pair and three-pair errors for BCH codes. First, we show the relation between the error-locator polynomials and the conflict-locator polynomial. Second, we show the relation among these polynomials and the syndromes. Finally, we provide how to correct the pair errors by solving equations including the relational expression by algebraic methods.