The trinity is both polytheistic and monotheistic. In Islam, there is not this problem, because there is no god but Allah. But Christianity is incoherent because of the trinity.
Monotheism: the doctrine of belief that there is but one God.
Polytheism: belief in or worship of more than one god.
According to the first plank of the doctrine of the trinity, it is Monotheistic in every sense of the word, as Tim says. However, according to the second and third plank, the trinity is Polytheistic in every sense of the word (Father god, Son god, Holy Spirit god. Each distinct.) That is three. Three is more than one.
To accept Christianity is to accept the trinity. To accept the trinity is to accept Polytheism and Monotheism, at the same time, which is to reject Polytheism and Monotheism at the same time. To accept the trinity is to reject the trinity. To accept Christianity is to reject Christianity at the same time.
The bible clearly teaches that there is only one God, and that there is only one person that is God. Jesus Christ was the word made flesh! The express image of God. Hebrews 1:3 The trinitarian view is incorrect and was devised years after the bible was written. There are three manifestations of God. Manifestations are different ways God deals with mankind... He deals with us as a Father, as a Son, and as the eternal Spirit. These three are all one person! The bible calls these manifestations: the Godhead. For in him dwelleth all the fulness of the Godhead bodily. (Colossians 2:9) monotheism!
Ward, your "manifestations" interact with each other in a personal way, demonstrating their separate personhood.
The Son obeys the Father (Philippians 2:8); the Son asks the Father for things (John 14:16); the Father loves the Son and gives Him all things (John 3:35; Luke 22:29); the Son honors the Father (John 8:49); the Son prays to the Father (John 17:1); the Father sends the Son (John 5:37, 10:36); the Son sends the Holy Spirit and the Holy Spirit testifies of the Son (John 15:26), the Spirit reminds Christ's followers of what the Son said (John 14:26), etc.
Call them "manifestations" if you like. They are still each uniquely personal.