Why and how does Quran use the words “the book”;

Bible in Koine Greek τὰ βιβλία ta biblia means “the books”…

Quran uses the word “The Book” several times when referring to Bible… It also contains connotations regarding Bible to be manmade and distinct from Torah and the Gospel which were the revelations from God…

[2:79] Therefore, woe to those who distort the book with their own hands, then say, "This is what GOD has revealed," seeking a cheap material gain. Woe to them for such distortions and woe to them for their illicit gains.

The above verse is talking about the Bible which is manmade but the leaders insist that it is from God.

[2:101] Now that a messenger from GOD has come to them and even though he confirms what they have, some followers of the book (Jews and Christians) disregard GOD's book behind their backs, as if they never have known.

This above verse is talking about Quran which confirms the Torah and Gospel, whilst the people of the Bible .i.e. the book hides the Quran (God’s book) behind their backs as if they don’t know that the Quran is from God… As you can witness that is what is happening with Christians and Jews they do not want to believe that the Quran is from God.

[2:113] The Jews said, "The Christians have no basis," while the Christians said, "The Jews have no basis." Yet, both of them read the book. Such are the utterances of those who possess no knowledge. GOD will judge them on the Day of Resurrection, regarding their disputes.

This (2:113) is mocking the Bible and saying that the Bible is telling lies to both Christians and Jews about their status as believers, because Bible is manmade and not from God.

[2:121] Those who received the book, and know it as it should be known, will believe in this. As for those who disbelieve, they are the losers.

(2:121) is once again talking about Bible but this time the Bible from God which is the Torah and the Gospel. Thus the emphasis on ‘… know it as it should be known …’ this happens when the Bible is used in conjunction with the Quran. And ‘… will believe in this …’ refer to (2:120) ‘… GOD’s, guidance is the real guidance …’ which is not the same as what their idols teach them.

There is lot more in the Quran about “the book” reference to Bible, which one can learn about and gain new knowledge from. Because the words “the book” in the Quran is very widely used my attempt to include all these instances would be very laborious and time consuming which cannot fit in this one article. Thus I leave this matter for the believers to investigate and use when and where it is suitable.