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So, I have been watching quite a bit of 古代 dramas。 I will say most if not all of them have the situation where this phrase comes up.
Does this phrase exist in the dramas only or did this actually happen in ancient China where usage of this phrase were applicable?
斬草除根 means killing a person and all his descendants so there is no chance for a revenge. Did it actually happen? Yes. It did happen and it's not the worst. 誅三族 means killing all of one's relatives of father's, mother's and wife's sides. In Qin dynasty, 李斯 received this savage punishment of 誅三族, which was ordered by 趙高, who, later, received the exact same punishment.
Worse than 誅三族 are 誅五族、誅七族、誅九族. The worst of all is 誅十族 which happened in Ming Dynasty to 方孝儒, a famous scholar。Besides all of his relatives, even his friends and students were killed. According to wikipedia, a total of 837 people were killed and thousands received other punishments.
斬草除根，常用的成語，用來敘述查明問題根源並斷絕後續可能發生的問題，其實在現代也是蠻常用的，語氣跟語意是比較強烈的一句成語。
例如：
a) 如果這個小問題不斬草除根，未來一定會造成嚴重的後果。
b) 預防蚊蠅孳生最斬草除根的作法，就是隨時注意環境的乾淨整潔！
c) (on tv Show)
A: 王小明知道太多了，盡快找到他！
B: 是！
A: 記得，務必斬草除根！(in this dialog, A means "kill 王小明".)
最後送OP一句算俏皮話吧，「斬草不除根，春風吹又生」。
斬草除根 is an existing idiom
典源：
見惡如農夫之務去草焉... 絕其本根，勿使能殖 (when you see evil, treat it like farmer treating weed... cut off from the root, not let it grow)
~
典故說明：
據《左傳．隱公六年》載，春秋時，結鄰的陳、鄭二國時常發生爭端而彼此交惡。後來鄭國曾提出請求，希望能夠與陳國和平共處。當時，陳桓公弟五父認為廣結善緣是國家的重要施政方針，因此建議接受講和。但陳桓公卻認為宋、衛兩國才是難以對付的敵手，鄭國算不了什麼，就拒絕接受鄭國的講和。結果在陳桓公六年五月時，鄭國入侵陳國，陳國大敗，被擄去許多財物及俘虜。導致這種結果的原因，就是因為陳桓公沒有廣結善緣，讓惡因滋長蔓生而不知悔改，隨之而來的便是自取禍害。周朝大夫周任曾說：「治理國家的，看到惡人惡事，就要像農民剷除田裡的雜草一樣，直接將聚生的雜草連根拔起，不讓它有再生的機會，這樣除去惡的，善的才能發揚。後來「斬草除根」這句成語就從這裡演變而出，用來比喻除去禍根，不留後患。」(used as a metaphor for "to thoroughly remove so there will not be future troubles)
