If all three of the words translated "divorce" truly mean "divorce", as we know it... then Jesus did teach that both the divorcer, and the innocent party, commit adultery when they remarry (Matthew 19:3-9).
However both the law given by Moses (Deuteronomy 24:1-4), and the instructions of Paul (I Corinthians 7:27-28), teach that remarriage is not sinful.
The true understanding of Scripture will teach one truth, and not contradict itself.
Several words from the Greek have been translated and understood to mean "divorce."
But only two of these two words truly refers to "divorce" as we know it.
The other word referred to what we would call- "separation."
The word that means "divorce" is lusin, meaning "unbound, released."
It is used by Paul in I Corinthians 7. It comes from the Greek word -"luo".
When the document used to show the person had gone through the Scriptural method of validating the reason for the divorce and the giving of the wife a double dowry, it was called a "bill of divorcement."
This is the word - "apostasion". It means literally, "having been departed or dismissed."
The other word is apolusa, which means "separated, has sent away." It also comes from the Greek word "luo" as did "lusin" as used by Paul, but it has a prefix added to it which alters it from "lusin." Notice that "communicate" differs from "excommunicate" because of a prefix, and "operation" is far different from "cooperation," "ordinate" from "coordinate," "door" from "outdoor," and "wit" from "outwit." Too many have assumed that since the root word is the same, the meaning was the same. It is not the same meaning, as is evident from usage in Scripture.
It is possible to be "sent away" (apolusa), without a divorce (lusin). Just as many couples today may separate and not divorce. For one of the two separated partners to remarry would be adulterous. A divorce is necessary to truly allow the former partners to remarry.
If we translate all these words as "divorce", we will find two teachings in Scripture that are unharmonious. This will allow the debate to go on, with each side choosing which verses they want to quote. We should be able to quote any verse, and no verse should contradict the correct teaching. We must never accept any interpretation of a Scripture that does not fit in perfect harmony with the rest of the Word of God.
The proper understanding of the words is best illustrated with this order:
First, in the process of the divorce, a man had to decide to release his wife-- lusin.
Secondly, he must get a bill of divorcement - apostasion - proving that he had presented the reason for divorce, and given the double dowry.
Thirdly, he could separate from her, and send her out of the home - apolusa.
 
This order, with all three steps was considered by God to be the real "divorce."
Lusin, initiated the process, apostasion presented the reason and gave the dowry so the woman was not destitute and unable to survive, and apolusa sent the partner from the home completing the process.
This order needed to be followed. God did not want the men to skip the first two steps and simply send the woman out of the home destitute.
With the importance of finding a harmonious and correct view on the subject, let's look at the Scriptures and examine the usage of these two words.
I Corinthians 7:27-28
"Art thou bound (married) unto a wife? seek not to be loosed (lusin=divorced). Art thou loosed (lusin -divorced) from a wife, seek not a wife. But if thou marry, thou hast not sinned..."
This verse shows that divorce and remarriage are discouraged, but if one is divorced and remarries, there is not sin.
This then brings agreement with the verses in Deuteronomy 24:1-2, which says that when a woman is given a bill of divorcement, she "may go and be another man's wife."
The purpose of Deuteronomy 24:1-2, was to show that the bill of divorcement made one free to remarry. If she were made to separate from her husband without a bill of divorcement, she would have then been committing adultery if she married another man.
Matthew 19:3,7-9
"Is it lawful for a man to put away (apolusa =separate from, not divorce) his wife, for every cause?... They say unto Him, 'Why did Moses command to give a writing of divorcement (apostasion = divorce) and to put her away (apolusa = separate from)? He saith unto them, 'Moses, because of the hardness of your hearts suffered (permitted) you to put away (apolusa = separate from) your wives, but from the beginning, it was not so. And I say unto you, Whosoever shall put away (apolusa = separate from) his wife, except it be for fornication, and shall marry another, committeth adultery, and whoso marries her that is put away (apolusa =separated from, not divorced) doth commit adultery."
The proper translation shows that this teaching does not disagree with I Corinthians 7 or Deuteronomy 24. Paul had taught that remarriage is not sinful after divorce and Jesus is not contradicting that. Jesus is answering the trick question of the Pharisees. They wanted to get Jesus on record on the practice of many Jewish men of skirting the law on divorce and simply sending their wives from the home. In this way, they avoided the requirement of the bill of divorcement of paying double dowry back to the wife.
Jesus is not fooled by their question. He gives a direct answer to them. He says that if one separates himself from his spouse without a bill of divorcement, he is committing adultery when he remarries (for he is not divorced in God's eyes). The wife is being forced to remarry and thereby also committing adultery. Her real husband has neglected to give her the bill of divorcement and the double dowry so she can provide for her needs.
"Fornication" is the only exception given to the need of the proper divorce. Since the fornication would have already have broken the marriage in God's eyes (and required stoning in the Old Testament), Jesus does not require the financial arrangements that the divorce would have required. In that case, Jesus did not condemn the Roman style divorce that many Jewish men were using.
Luke 16:18
"Whosoever putteth away (apolusa = separates from, not divorces) his wife, and marries another, commits adultery; and whosoever marries her that is put away (apolusa = separated from) her husband, committeth adultery."
Matthew 5:31
"It hath been said, 'Whosoever shall put away (apolusa = separate from) his wife, let him give her a writing of divorcement (apostasion =divorce). But I say unto you, that whosoever shall put away (apolusa = separate from) his wife, save for the cause of fornication, causes her to commit adultery; and whosoever shall marry her that is divorced (apolusa = separated from) committeth adultery.'"
Separation is not to be tolerated. If the marriage is not to be held together, a bill of divorcement must be given. Separation and remarriage is adultery. Divorce and remarriage is not adultery and Deuteronomy 24 and I Corinthians 7 agree with the teachings of Jesus.
Fornication made it unnecessary for the man to make sure the wife had a fair financial settlement. The fornication had broken the marriage in God's eyes. Any other cause for divorce requires for a man to make sure that the wife is not going to be sent out without financial ability to survive.
The fact that remarriage of a divorced person is not sinful, should not encourage divorce. Divorce should be recognized as being usually sinful. Yet, it is not an unpardonable sin. Any who have been divorced and remarried, should be encouraged to look back at the failed marriage and repent of any sins the Holy Spirit reveals. Once this is done, the remarried person has no reason or Scripture to prevent them from serving on the board, or from pastoring a church.
Matthew 19:6
"What therefore God hath joined together, let not man put asunder (this is another word, 'choridzo')."
This word - choridzo - means "to separate, to divide." Therefore in a marriage, even a mere separation is to be discouraged. God does not even want separation between married ones, let alone divorce. But after divorce does occur, God can and will forgive the repentant. To remarry is not sinful, and Christendom must forgive as God forgives, and not stigmatize those who have remarried.
This is in agreement with Paul's teaching in I Corinthians 7:10-12:
"Let not the wife depart (apolusa = separate from) her husband, but if she depart (apolusa = separate from), let her remain unmarried... or be reconciled... let not the husband put away (apolusa = separate from) his wife..."