Of the approximately 264 thousand words in the Book of Mormon, about 17 thousand are close parallels to the King James translation of the Bible. Most parallel verses occurred when Nephi quoted the Isaiah of the Old Testament using records on brass plates brought from Jerusalem. Other parallels occurred when the resurrected Savior repeated his Sermon on the Mount to the Nephites and then quoted Malachi at length. In both cases, we are told in the text that these are quotations of scriptures that had been recorded elsewhere.
Joseph Smith left no record of how he translated the plates beyond saying that it was done by the power of God. LDS scholars generally agree that in instances where the Book of Mormon parallels the Bible, Joseph Smith must have noted the parallels and used the King James Bible to guide him in his choice of words. If the Book of Mormon agreed with the Biblical text in meaning, he apparently utilized the Biblical text, italicized words and all. However, when the plates differed from the Biblical text, he followed the text on the plates.
For example, of 433 verses of Isaiah quoted in the Book of Mormon, 46 percent are identical to those in the Bible, while 54 percent are modified to some extent. These different verses have been of great use to LDS scholars. Several Hebrew literary structures are found only in a degraded form in the King James Bible, while those literary structures are complete and intact in the Book of Mormon. This shows that the brass plates version found in the Book of Mormon was a different, older Isaiah text.
A slightly more difficult problem is posed by Book of Mormon similarities with Biblical scriptures which were not in existence when Lehi left Jerusalem. For instance, Moroni 7:45, 46 parallels 1 Corinthians 13:4-8 (Paul's discussion on charity) so closely that some explanation is called for.
Mormon, Moroni's father, lived after Christ and knew Christ's teachings. Since the ultimate source of Paul's teachings was also Christ, it should not be surprising to find both Mormon and Paul teaching the same thing. The fact that both wrote on charity indicates the importance of this gospel concept.
If Mormon delivered a sermon on charity, it could logically contain much of the same material as Paul's teachings on the same subject. However, one would not expect identical, word-for-word renderings of the two writings, and indeed they are not totally identical. Two significant clauses found in I Corinthians are absent from Mormon's sermon, and Mormon's definition of charity is absent from Paul's writings. Nonetheless, several phrases are identical. These identical phrases are understood the same way as the Isaiah passages by Latter-day Saints: Joseph Smith recognized Mormon's teachings as similar to Paul's and likely used the biblical text where it agreed in meaning with the plates. Where the meanings differed, he followed the plates.
Since the ultimate source of the teachings of the Book of Mormon is Christ, and since the ultimate source of the teachings of the Bible (both Old and New Testaments) is also Christ, it should surprise no one that there are many parallels between Book of Mormon and Bible passages. While one cannot conclusively prove that Joseph Smith used a King James Bible as an aid in translating the parallel passages, that explanation is reasonable.