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1835 Doctrine and Covenants denies polygamy
Doctrinal foundation of plural marriage
Introduction of plural marriage
Questions about Joseph Smith and plural marriage
Notable plural wives of Joseph Smith
Plural marriage in Utah
End of plural marriage
The Doctrine and Covenants
1835 Doctrine and Covenants denies polygamy
Question: Why did the 1835 Doctrine and Covenants include a statement of marriage that denied the practice of polygamy at a time when some were actually practicing it?
Polygamy was not being taught to the general Church membership at that time
The Article on Marriage was printed in the 1835 D&C as section 101 and in the 1844 D&C as section 109. The portion of the Article on Marriage relevant to polygamy states:
Inasmuch as this church of Christ has been reproached with the crime of fornication, and polygamy: we declare that we believe, that one man should have one wife; and one woman, but one husband, except in case of death, when either is at liberty to marry again. 
This was true—the Church membership generally was not being taught plural marriage, and were not living it at that time.
The statement itself was not changed between the 1835 and 1844 editions of the D&C
In fact, the statement remained in the D&C until the 1876 edition, even though plural marriage had been taught to specific individuals since at least 1831, practiced in secret since 1836, and practiced openly since 1852. The matter of not removing it in 1852 was simply due to the fact that a new edition of the D&C was not published until 1876.
The available evidence suggests that Joseph Smith supported its publication
While some have suggested that the article was published against Joseph's wishes or without his knowledge, the available evidence suggests that he supported its publication. It was likely included to counter the perception that the Mormon's practice of communal property (the "law of consecration") included a community of wives.
The statement was not a revelation given to Joseph Smith - it was written by Oliver Cowdery
This statement was not a revelation given to Joseph Smith—it was written by Oliver Cowdery and introduced to a conference of the priesthood at Kirtland on 17 August 1835. Cowdery also wrote a statement of belief on government that has been retained in our current edition of the D&C as section 134. Both were sustained at the conference and included in the 1835 D&C, which was already at the press and ready to be published. Joseph Smith was preaching in Michigan at the time Oliver and W.W. Phelps introduced these two articles to the conference; it is not known if he approved of their addition to the D&C at the time, although he did retain them in the 1844 Nauvoo edition, which argues that he was not opposed to them. (Phelps read the article on marriage, while Cowdery read the one on government.) 
Some have suggested that the manner in which the conference was called suggests that Joseph was not the instigator of it, since it seems to have been done quite quickly, with relatively few high church leaders in attendance:
The General Assembly, which may have been announced on only twenty-four hours' notice, was held Monday, August 17[, 1835]. Its spur-of-the-moment nature is demonstrated by observing that a puzzling majority of Church leaders were absent. Missing from the meeting were all of the Twelve Apostles, eight of the twelve Kirtland High Council members nine of the twelve Missouri High Council members, three of the seven Presidents of the Quorum of Seventy, Presiding Bishop Partridge, and...two of the three members of the First Presidency. 
However, there is also some evidence that an article on marriage was already anticipated, and cited four times in the new D&C's index, which was prepared under Joseph's direction and probably available prior to his departure. Thus, "if a disagreement existed, it was resolved before the Prophet left for Pontiac." 
On July 7, 1878, Joseph F. Smith discussed Oliver's awareness of polygamy at the time of this publication:
To put this matter more correctly before you, I here declare that the principle of plural marriage was not first revealed on the 12th day of July, 1843. It was written for the first time on that date, but it had been revealed to the Prophet many years before that, perhaps as early as 1832. About this time, or subsequently, Joseph, the Prophet, intrusted this fact to Oliver Cowdery; he abused the confidence imposed in him, and brought reproach upon himself, and thereby upon the church by "running before he was sent," and "taking liberties without license," so to speak, hence the publication, by O. Cowdery, about this time, of an article on marriage, which was carefully worded, and afterwards found its way into the Doctrine and Covenants without authority. This article explains itself to those who understand the facts, and is an indisputable evidence of the early existence of the knowledge of the principle of patriarchal marriage by the Prophet Joseph, and also by Oliver Cowdery. 
However, there continues to be debate about whether Oliver Cowdery knew about--or prematurely practiced--plural marriage in the 1830s.  Oliver would learn about the Fanny Alger marriage, but his reaction at the time seems to have been wholly negative.
The original D&C 101 article outlined the general practice of performing a Latter-day Saint wedding, explained LDS beliefs about the marriage relationship, and denied that the Saints were practicing polygamy.
Question: Was the practice of polygamy general knowledge among Latter-day Saints in 1835 when the "Article on Marriage" was published?
Knowledge of the practice of polygamy among the Saints was limited prior to the 1840s
Some have argued that rumors of "polygamy" may already have been circulating as a result of the Prophet teaching the concept to some of his close associates. However, Brian Hales has argued that there are few if any extant attacks on Joseph or the Saints about polygamy prior to the 1840s:
...if the article was designed to neutralize reports about Joseph Smith and his alleged "crimes," polygamy would not have been included because that allegation was not made then nor at any other time during the Kirtland period according to any documentation currently available. In other words, assuming that the denial of polygamy in the "Marriage" article [of D&C 101] was specifically tied to rumors of Joseph Smith's behavior is problematic, unless other corroborating evidence can be located. 
On the other hand, charges of polygamy or "free love" or having wives in common were often made against new or little-known religious or social groups. As Hales reports:
Some [nineteenth-century utopian societies] experimented with novel marital and sexual practices, which focused suspicion on all the groups....Accordingly, early Latter-day Saint efforts to live the law of consecration, even though it sustained traditional monogamy, were instantly misunderstood....
John L. Brooke...wrote: "Among the non-Mormons in Ohio there were suspicions that the community of property dictated in the 'Law of Consecration' included wives."...
It seems plausible, even likely, that beginning in 1831, some uninformed individuals assumed that the law of consecration included a community of wives as one of its tenets, even publishing such claims, although there is no indication that this is how the Mormons themselves interpreted the law of consecration. Understandably, Church leaders would actively seek to deny such untrue allegations in a document on marriage to be included in the 1835 Doctrine and Covenants. 
Gilbert Scharffs notes:
The original Section 101 (never claimed as a revelation but approved as a statement of belief) did state that monogamy was the practice of the Church at that time. The section was not written by Joseph Smith and was voted upon by members in his absence. Perhaps the section was intended to prevent members from getting involved with plural marriage until such a time as the practice would be authorized by the Lord Church-wide. When that became the fact, the current Section 132 replaced the old Section 101. 
- Doctrine and Covenants, 1835 edition, Section 101.
- History of the Church, 2:246–247. Volume 2 link
- Brian C. Hales, Joseph Smith's Polygamy Volume 1: History (Salt Lake City, Utah: Greg Kofford Books, 2013), 154.
- Hales, Joseph Smith's Polygamy Vol. 1, 173, see pp. 171–1731 for full details.
- Joseph F. Smith, Journal of Discourses 20:29.
- Hales, Joseph Smith's Polygamy Vol. 1, 156–158.
- Hales, Joseph Smith's Polygamy Vol. 1, 161–162.
- Hales, Joseph Smith's Polygamy Vol. 1, 166, 168.
- Gilbert Scharffs, "Marriage Is Ordained of God", The Truth About "The God Makers" off-site