There is not a single verse from the New Testament that prohibits polygamy. Christians usually mistakenly present the following verses from the Bible to prove that polygamy in the New Testament is not allowed:
Matthew 19:1-12 "1. When Jesus had finished saying these things, he left Galilee and went into the region of Judea to
the other side of the Jordan.
2. Large crowds followed him, and he healed them there.
3. Some Pharisees came to him to test him. They asked, "Is it lawful for a man to divorce his wife for any and every reason?"
4. "Haven't you read," he (Jesus) replied, "that at the beginning the Creator `made them male and female,'
5. and said, `For this reason a man will leave his father and mother and be united to his wife, and the two will become one flesh' ?
6. So they are no longer two, but one. Therefore what God has joined together, let man not separate."
7. "Why then," they asked, "did Moses command that a man give his wife a certificate of divorce and send her away?"
8. Jesus replied, "Moses permitted you to divorce your wives because your hearts were hard. But it was not this way from the beginning.
9. I tell you that anyone who divorces his wife, except for marital unfaithfulness, and marries another woman commits adultery."
10. The disciples said to him, "If this is the situation between a husband and wife, it is better not to marry."
11. Jesus replied, "Not everyone can accept this word, but only those to whom it has been given.
12. For some are eunuchs because they were born that way; others were made that way by men; and others have renounced marriage because of the kingdom of heaven. The one who can accept this should accept it."
In the above verses, we see that Jesus was approached with a question about whether or not it is allowed for a man to divorce his wife in Matthew 19:3. Jesus immediately referred to the Old Testament for the answer in Matthew 19:4. He referred to Adam and Eve, one man and one woman. The Old Testament does talk about the story of Adam and Eve as one husband and one wife. However, the Old Testament which Jesus had referred to in Matthew 19:3 does allow polygamy.
Also, when a man becomes a one flesh with his wife in Matthew 19:5-6, this doesn't mean that the man can't be one flesh with another woman. He can be one flesh with his first wife, and one flesh with his second wife, and one flesh with his third wife and so on.... To further prove this point, let us look at the following from the New Testament:
Matthew 22:23-32 "23. That same day the Sadducees, who say there is no resurrection, came to him with a question.
24. "Teacher," they said, "Moses told us that if a man dies without having children, his brother must marry the widow and have children for him.
25. Now there were seven brothers among us. The first one married and died, and since he had no children, he left his wife to his brother.
26. The same thing happened to the second and third brother, right on down to the seventh.
27. Finally, the woman died.
28. Now then, at the resurrection, whose wife will she be of the seven, since all of them were married to her?"
29. Jesus replied, "You are in error because you do not know the Scriptures or the power of God.
30. At the resurrection people will neither marry nor be given in marriage; they will be like the angels in heaven.
31. But about the resurrection of the dead--have you not read what God said to you,
32. `I am the God of Abraham, the God of Isaac, and the God of Jacob' ? He is not the God of the dead but of the living."
In Matthew 22:24-28, the Jews referred to Deuteronomy 25:5 from the Old Testament where it states that if a woman's husband dies, and she didn't have any kids from him, then she must marry his brother regardless whether he had a wife or not. When the Jews brought this situation up to Jesus in Matthew 22:24-28, Jesus did not prohibit at all for the childless widow to marry her husband's brother (even if he were married). Instead, Jesus replied to them by saying that we do not marry in heaven, and we will be like angels in heaven (Matthew 22:30).
So in other words, if Jesus allowed for a widow to marry her former husband's brother even if he were married, then this negates the Christians' claim about the Bible prohibiting polygamy. A man can be one flesh with more than one woman. In the case of Matthew 22:24-28, the man can be one flesh with his wife, and one flesh with his deceased brother's wife. Also keep in mind that Exodus 21:10 allows a man to marry an infinite amount of women, and Deuteronomy 21:15 allows a man to marry more than one wife.