An ill-timed run of four successive defeats heading into today's match meant that the Tangerines' hopes of survival always looked bleak, needing a win, a Ross County defeat and an eight-goal swing in goal difference in order to escape bottom spot.
A miracle was still just about on the cards around the hour mark of both games, with Dundee United winning and Ross County losing by two, although a further five-goal swing was still needed.
There was to be no such drama, though, with Motherwell coming from behind to win 3-2, making Ross County's 3-1 defeat to Kilmarnock irrelevant.
Dundee United got off to the worst possible start in their unlikely bid for survival, with Kevin van Veen giving Motherwell the lead after only five minutes.
However, a Jamie McGrath penalty on the half-hour mark pulled them level, before Steven Fletcher raised hope by putting them in front four minutes into the second half.
That hope was fuelled by news that Ross County had fallen two goals down in the very same minute, but Yan Dhanda then pulled one back for County from the spot to make the required swing in goal difference even less likely.
As it was, Dundee United could not hold up their end of the bargain either, with goals from Blair Spittal and Max Johnston for Motherwell in the final 20 minutes to earn them a 3-2 victory.
The result sees the Terrors drop into the second tier after three seasons in the top flight, while Ross County will now go into the Scottish Premiership playoffs to determine their survival.