Why does it say "va'Yiva'etz"?
Sanhedrin 101b: He put a Rasha next to each Tzadik and addressed the nation. He said, do you agree to everything I will do? Do you want me to be your king? Will you do everything I say? All said 'Yes!' Is this even if I tell you to serve idolatry? The Tzadik said Heaven forbid! The Rasha said, surely, Yarav'am would not serve idolatry. He is merely testing us!
Radak: He took counsel; it agreed with what he himself thought.
Why did he make golden calves?
Radak: He enticed Yisrael with words. He said, Hashem wanted to split the kingship, like Achiyah ha'Shiloni ha'Navi told me. Hashem does not want Malchus Beis David, and not Yerushalayim, which is [the capitol of] Malchus Beis David. If so, we should make another place to offer Korbanos. He said that Aharon made an Egel to cause the Shechinah to dwell in place of Moshe. Also now that you do not have a place of Shechinah, i.e. Yerushalayim, we will make an Egel in place of it to bring down Shechinah. He said "Hinei Elokecha Yisrael Asher He'elucha me'Eretz Mitzrayim" like they said about the Egel in the Midbar (Shemos 32:8). They did not intend for idolatry.
Malbim: He could have build a Mikdash is Beis Kel like the Beis ha'Mikdash in Yerushalayim, but it was difficult in his eyes. Shlomo build the Beis ha'Mikdash amidst extraordinary wealth; Yaravam could not build a house like it. If he would build it without glory and honor, people would go to the great house [in Yerushalayim], also because it was known that Hashem dwells there, from the fire from Shamayim that was always there. It had also the Aron and Moshe's Kelim, and people saw that Kevod Hashem filled it at the time it was built. All this would be missing from any house he built. Therefore, instead he made calves. His main purpose was to deter people from Avodas ha'Mikdash. Surely he showed good facets of it from all angles. To Yir'ei Hashem, he said that they are l'Shem Hashem; just like He dwells on the Keruvim, which are from the Merkavah, which is a human form that represents Chachmah, he enticed them to say that He will dwell in Beis Kel and in Dan, due to the face of the ox on the Merkavah, which represents great harvests and the strength of an ox. The verses show that he made Kohanei Bamos only later. He thought that just like the nation agreed to cast off the heavy yoke of a king and officers -this is why they rebelled against Shlomo 1 and chose him, for he would lighten the yoke of kingship - they would also want to cast off the yoke of Avodas Beis Hashem. Also it was a burden on them - going three times a year to a far place, and many Korbanos for every sin 2 , voluntary offerings, Terumos, Ma'aseros 3 etc. The Egel represents freedom from all Avodah 4 . This would exempt them from the yoke of Avodas ha'Mikdash and all Mitzvos!
Malbim citing Mahari They were not l'Shem Avodah. 5 Yisrael would not consent. They knew that Hashem was angry about the Egel in the Midbar, and sent a plague! Rather, it was a commemoration of his kingship. He was from Shevet Yosef; its Siman was an ox - "Bechor Shoro Hadar Lo" (Devarim 33:17).
Only Yaravam rebelled against Shlomo. Yisrael rebelled only against Rechav'am! Perhaps Malbim means that they resented Shlomo and wanted to rebel, but they were afraid, and did not rebel in action until he died and his son wanted to continue in his ways. (PF)
The Torah explicitly obligates only for sins of Kares (and a few others, for which one brings Asham or Oleh v'Yored)! However, for Bitul Aseh, Lav ha'Nitak l'Aseh and improper thoughts, one should bring an Olah to atone.
What is the source that Yaravam annulled Terumos and Ma'aseros? Perhaps primarily he wanted to annul eating Ma'aser Sheni in Yerushalayim; to do so, he totally exempted them from all Terumos and Ma'aseros. (PF)
Sanhedrin 63b: Yisrael served [the Egel] only in order to permit Arayos. Rashi (63b) - they desired only Arayos. They totally cast off the yoke, so people would not rebuke them about Arayos.
Malbim: It says "Lezabea'ach la'Agalim" (verse 32), and "Asah Lachem Yaravam lei'Lohim" (Divrei ha'Yamim II, 13:8).
Why did he make two calves?
Radak: He did not want to exert Yisrael (his 10 Shevatim) to come to one place.
Malbim citing Mahari: They were a commemoration of his kingship. He was from Shevet Yosef, which consists of Efrayim and Menasheh.
Malbim: They saw in Egypt two bulls that they served - one in the capitol Mamafis (Mof), and the other in Halifalis (Beis Shemesh). The festival for the Egel was in the eighth month, for the man for whom the Egel was made (Sarafis or Aziras), that he and his wife taught agriculture, was killed in that month. They honored the ox in Mof very much and spent a fortune on it, therefore the Egel in Beis Kel 1 was primary.
I.e. it corresponded to the bull in Mof. (PF)
Why did he say "Rav Lachem me'Alos Yerushalayim"?
Radak: It is enough that you went until now.
Radak citing Targum Yonasan: It is a long way to ascend to Yerushalayim; these are your [deities] that you fear.
Radak citing a Midrash: He set up two guards or soldiers - one at Tavor, and one at ha'Mitzpah [to prevent people from ascending], like it says in Hoshe'a (5:1) "Ki Pach Heyisem l'Mitzpah v'Reshes Perusah Al Tavor."
Malbim: It is a big toil to ascend, both due to the distance, and the cost of Ma'aseros and Korbanos.
Why did he say "Hinei Elokecha Yisrael Asher He'elucha me'Eretz Mitzrayim"?
Malbim: Hashem took you from being slaves. You are free from all Avodah - both to Hashem and to people.