Perhaps.. I would have to re-read those passages (a much wider context than many would do). But as to the Reason that the "United Kingdom of Israel" was divided was because of an action of King David.. the consequence of his action coming by prophecy that the Kingdom would be divided.

So now I go a refreshing of the topic as to When that united Kingdom was divided as to whether or not it was AT the death of Solomon, or before, or sometime but much later after...

In most of the Scripture the "Northern Alliance" (what i have come to call it) is usually called, "the Kingdom of Israel" in contrast to its usage of the phrase "the Kingdom of Judah".

When one comes to the phrase "kingdom of Israel" in the Scriptures one does have to play close attention to the context including the times... Much like the term "Israel" one cannot knee jerk or trigger that it means a particular "Israel" one should take time for figure out just which "Israel" is being referred to. I am sure you take that time, many don't and this brings wider confusion rather than clarification.