See this passage explained in Matthew 16:1-12.
Mark 8:12
Sighed deeply in his spirit - His heart was deeply affected at their wickedness and hypocrisy. The word “spirit” here is taken as the seat of the emotions, passions, affections. He drew groans deeply from his breast.
No sign be given - That is, no such sign as they asked, to wit, a sign “from heaven.” He said a sign should be given, the same as was furnished by Jonas, Matthew 16:4. But this was not what they “asked,” nor would it be given “because” they asked it.
Mark 8:15
Beware of the leaven of the Pharisees - See Matthew 16:6.
Of Herod - Of the Herodians - of Herod and his followers. Matthew, instead of “Herod,” has “the Sadducees.” It is not improbably that he cautioned them against them all. The Pharisees sought his life, and were exceedingly corrupt in their doctrine and practice; the Sadducees denied some of the essential doctrines of religion, and the Herodians probably were distinguished for irreligion, sensuality, and corrupt living. They were united, therefore, with the Pharisees and Sadducees in opposing the claims of Jesus. Matthew has recorded his caution to avoid the Pharisees and Sadducees, and Mark has added, what Matthew had omitted. the caution likewise to beware of the Herodians. Thus, the evangelists speak the same thing.
Then taking a boat with His disciples, He crossed the lake to Magdala, at the southern end of the plain of Gennesaret. In the border of Tyre and Sidon His spirit had been refreshed by the confiding trust of the Syrophoenician woman. The heathen people of Decapolis had received Him with gladness. Now as He landed once more in Galilee, where His power had been most strikingly manifested, where most of His works of mercy had been performed, and His teaching given, He was met with contemptuous unbelief. DA 405.1
A deputation of Pharisees had been joined by representatives from the rich and lordly Sadducees, the party of the priests, the skeptics and aristocracy of the nation. The two sects had been at bitter enmity. The Sadducees courted the favor of the ruling power in order to maintain their own position and authority. The Pharisees, on the other hand, fostered the popular hatred against the Romans, longing for the time when they could throw off the yoke of the conqueror. But Pharisee and Sadducee now united against Christ. Like seeks like; and evil, wherever it exists, leagues with evil for the destruction of the good. DA 405.2
Now the Pharisees and Sadducees came to Christ, asking for a sign from heaven. When in the days of Joshua Israel went out to battle with the Canaanites at Bethhoron, the sun had stood still at the leader's command until victory was gained; and many similar wonders had been manifest in their history. Some such sign was demanded of Jesus. But these signs were not what the Jews needed. No mere external evidence could benefit them. What they needed was not intellectual enlightenment, but spiritual renovation. DA 406.1
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