Answer: Glad your shrub has adjusted to its new home and is blooming beautifully. It isn't necessary to remove the spent flowers, but you certainly can. In fact, clipping them off at the base will encourage additional flowering stems. You can trim a few inches of stem as you cut the old flowers off. This will keep the plant neat and trim, and you'll have substantial new growth throughout the season.
It is normal for some of the oldest leaves to turn yellow and fall off. These are the leaves near the main stem of the plant, plus those nearest the bottom. Its the plant's way of renewing itself and new leaves will be produced where the old leaves once were.
Enjoy your gardenia!
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