The miraculous victory of Badr greatly troubled the idolators and the Jews of Medina. They felt that their position was now degenerating into one of weakness and isolation. Only 2 years after his migration, the Prophet of God had managed to brake the traditional pattern of power distribution in the desert. The enemies of Islam would meet clandestinely and encourage the composition and recitation of divisive poetry.
Ka'ab ibn Ashraf, a Jewish chieftain of Banu Nadhir, was a poet of considerable fame and he used to recite in the gatherings fiery poems inciting the people to rise up against Islam. This was a clear breeching of the Medina covenant of peace with the Muslims, non partisanship which eachother's enemies. ibn Ashraf's particularity as compared to the other non Muslims and hypocrites that secretly disliked Islam and conspired against it, is that he openly joined the Meccan ranks with whom the community was at war, becoming a propaganda tool that composed eulogies mourning the Meccan chiefs slain in the battle of Badr and defamed Muslim women.
The closest one can come to the kind of impact this kind of poetry had in Arabian tribal life in those days, is to remember the role propaganda played during the world wars of the 20th century, more particularly the 2nd one. The chief propagandists among the Nazis were regarded as top priority targets by Western authorities. The issue here is thus not that of low-level disparaging comments and mockeries, rather the kind of criticism with deadly ramifications. The Quran and hadith contain many instances of the prophet and the Muslims being the targets of mockery and ridicule, both in times of political weakness and strength, yet neither responded in kind nor retaliated violently. The Quran for instance in sura tawba relates how the Medina hypocrites would engage in injurious talk about the prophet, and this at a time where the Muslim community had become powerful. The only response they got from the prophet was that he socially ostracized them, refusing their charity donations, and leaving their fate to Allah in the Hereafter
9:66"If We pardon one faction of you - We will punish another faction because they were criminals".
As to Kaab, his animosity was such that it is said the verse 4:51 speaking of Jews believing in idols alludes to him, when he accompanied a delegation from Medina to Mecca in search of an alliance against the Muslims, and publicly bowed to the idols to reassure the suspecting Quraysh
"Your are people of Scripture and Muhammad has a Scripture and we are not completely sure that this is a scheme that you devised. So if you want us to go along with you, you have to prostrate to these two idols and believe in them".
But being a coward he never attended the battles himself, preferring to plot and incite behind closed doors. His role in galvanizing the Quraysh prior to the battle of Uhud is well known, his wife herself is reported to have warned him that his life was at threat because of his actions. Although the prophet said that Kaab was deserving of being put to death since he should be treated as a combatant, he nevertheless did not plan the execution. It is to be noted that any modern government seeking to preserve the survival of its people in times of war, would look to target specific opponents whose death would have a more significant impact in the long-run in terms of avoiding further bloodshed.
He was thus incited out of his hiding place and killed, which succesfully prevented an all out war with the Bani Nadir. Other opinions say his assassination occured after the battle of Uhud in response to an attempted murder of the prophet.
Further reading answering Sam Shamoun "Muhammad and the Ten Commandments"
