But he answered and said, It is not meet to take the children's bread, and to cast [it] to dogs.
Note 2 at Mt 15:26: It is very clear in Scripture that Jesus was sent to fulfill God's covenant to the Jews. However, it was equally clear that Jesus would open up faith unto the Gentiles (Ac 13:46-47, 15:13-18; and Ro 9:24-33). Jesus certainly knew this (Ac 15:18) and had already ministered to numerous Gentiles without the apparent disdain that we see here with the Syrophenician woman (Mt 8:5-13; Lu 7:1-10; Joh 4:9, and 39-42). Jesus could not have been forced into ministering to this woman's daughter if it was not His will to do so. Therefore, His silence and rough answer to this woman must have been designed to accomplish a positive result.
As mentioned in note 5 at Joh 2:5, humility is an important ingredient of faith. This woman was a stranger from the covenants of promise (Eph 2:12) and had no right to demand anything. Jesus' silence and then comparison of her to a dog would certainly have offended an arrogant person. It is possible that for this very reason, Jesus tried her faith. Jesus didn't need to do this with the centurion in Lu 7:6-7, because the centurion had already humbled himself. Compare this verse with the parable of the Pharisee and publican in Lu 18:9-14.