AND IF THE FIRST PIECE OF DOUGH BE HOLY THE LUMP IS ALSO: ei de e aparche hagia kai to phurama: (Exodus 22:29; 23:16,19; Leviticus 23:10; Deuteronomy 18:4; 26:10; Nehemiah 10:35-37; Proverbs 3:9; Ezekiel 44:30; James 1:18; Revelation 14:4):

The Amplified Version helps understand Paul's meaning translating this as...

Now if the first handful of dough offered as the firstfruits [Abraham and the patriarchs] is consecrated (holy), so is the whole mass [the nation of Israel]; and if the root [Abraham] is consecrated (holy), so are the branches.

The Bible Knowledge Commentary explains that...

Paul was convinced that Israel’s stumbling is temporary rather than permanent and that the nation will be restored as God’s people. With two illustrations Paul showed why he believed this. His first illustration was taken from God’s instructions to Israel to take “a cake from the first of [their] ground meal and present it as an offering” (Nu 15:20) after they entered the land of Canaan and reaped their first wheat harvest. This offering was to be repeated each year at their harvests. The cake made from the first ground meal of the wheat harvest was sanctified or made holy by being offered to God...Paul’s second illustration was that of a tree: If the root is holy, so are the branches. In both illustrations the principle is the same: what is considered first contributes its character to what is related to it. With a tree, the root obviously comes first and contributes the nature of that type of tree to the branches that come later. With the cake presented to the Lord, the flour for the cake is taken from the ground meal, but that cake is formed and baked first and presented as a firstfruit. Since it is set apart to the Lord first, it sanctifies the whole harvest. The firstfruits and the root represent the patriarchs of Israel or Abraham personally, and the lump and the branches represent the people of Israel. As a result Israel is set apart (holy) to God, and her “stumbling” (rejection of Christ) must therefore be temporary. (Walvoord, J. F., Zuck, R. B., & Dallas Theological Seminary. The Bible knowledge commentary: An exposition of the scriptures. Wheaton, IL: Victor Books)

First piece (536) (aparche from apó =  away from + árchomai = to begin) is the first fruit, which in Biblical terms describes an offering of any kind, animal as well as grain. It represents the first portion of offering set aside specifically for Lord. The first portion of the harvest was regarded both as a first installment and as a pledge of the final delivery of the whole. Here the first fruit and the lump speak of dough, not of fruit or grain.

Holy (40) (hagios) (Click for in depth analysis of hagios) means set apart from profane  common use by God and for God.

Paul is alluding to the practice described in Numbers 15 where we read...

Then the LORD spoke to Moses, saying, 18 "Speak to the sons of Israel, and say to them, 'When you enter the land where I bring you, 19 then it shall be, that when you eat of the food of the land, you shall lift up an offering to the LORD. 20 'Of the first of your dough you shall lift up a cake as an offering; as the offering of the threshing floor, so you shall lift it up. 21 'From the first of your dough you shall give to the LORD an offering throughout your generations. (Numbers 15:17-21)

William MacDonald explains that...

The argument is that if the piece of dough is set apart to the Lord, so is all the dough that might be made from it (MacDonald, W., and Farstad, A. Believer's Bible Commentary: Old and New Testaments. Nashville: Thomas Nelson)

J Vernon McGee writes that...

“Dough,” of course, is bread dough! A part of the dough was offered to God as a token that all of it was acceptable. The “firstfruit” evidently refers to the origin of the nation: Abraham, Isaac, and Jacob. “Holy” has no reference to any moral quality, but to the fact that it was set apart for God. Now if the first fruit, or the first dough—that little bit of dough—was set apart for God, what about the whole harvest? Since Abraham, Isaac, and Jacob were set apart for God, what about the nation? It all belongs to God, you see. God is not through with the nation Israel. (McGee, J. V.  Thru the Bible commentary. Based on the Thru the Bible radio program. Vol. 4, Page 724. Nashville: Thomas Nelson)

Warren Wiersbe adds that...

The reference is to (Numbers 15:17-21). The first part of the dough was to be offered up to God as a symbol that the entire lump belonged to Him. The same idea was involved in the Feast of Firstfruits, when the priest offered a sheaf to the Lord as a token that the entire harvest was His (Lev 23:9-14). The basic idea is that when God accepts the part He sanctifies the whole. Applying this to the history of Israel, we understand Paul’s argument. God accepted the founder of the nation, Abraham, and in so doing set apart his descendants as well. God also accepted the other patriarchs, Isaac and Jacob, in spite of their sins or failings. This means that God must accept the “rest of the lump”—the nation of Israel.

AND IF THE ROOT BE HOLY, THE BRANCHES ARE TOO: kai ei e rhiza hagia, kai oi kladoi: (17; Genesis 17:7; Jeremiah 2:21; 1 Corinthians 7:14)

Root = The patriarchs, Abraham, Isaac, and Jacob

Branches = The patriarchs’ descendants: the nation of Israel.

If the foundational part of a plant (the root) is holy, then that which it produces (the branches) must likewise (too) be holy.

Branches...broken off ~ unbelieving Israel

Wild Olive ~ believing Gentile

Rich root of the Olive Tree ~ The place of divine blessing. God’s covenant of salvation made with Abraham (John MacArthur)

Ryrie agrees that...

The olive tree is the place of privilege that was first occupied by the natural branches (the Jews). The wild branches are Gentiles who, because of the unbelief of Israel, now occupy the place of privilege. The root of the tree is the Abrahamic covenant that promised blessing to both Jew and Gentile through Christ. (Ryrie Study Bible)

If is a first class condition, which assumes that what follows is a fact. Indeed some of the branches, the natural Jews (Israel) were broken off by their unbelief and rejection of the Messiah. Notice that Paul makes clear that not all of the branches (the natural Jews) were broken off. This truth is also emphasized by his use of the phrase among them which refers to natural Jews who did believe in Messiah. There always had been a believing remnant in Israel.

Notice that the term “Broken off” is the equivalent of “fall” (Ro 11:11), “their failure” (Ro 11:12), and “their rejection” (Ro 11:15).

AND YOU BEING A WILD OLIVE WERE GRAFTED IN AMONG THEM: su de agrielaios on (PAPMSN) enekentristhes (2SAPI) en autois: (Psalms 80:11-16; Isaiah 6:13; 27:11; Ezekiel 15:6-8; Matthew 8:11,12; ; John 15:6) (Deuteronomy 8:8; Judges 9:8,9; Psalms 52:8; Zechariah 4:3; Jonah 1:16; Revelation 11:4)

Wild Olive (65) (agrielaios from ágrios = wild + elaía = olive tree, olive) is Paul's metaphorical way of referring to the Gentiles who had believed in Christ.

Centuries earlier God had warned His people through His prophet Jeremiah that their continued unbelief and idolatry would reap an unwanted harvest ...

What right has My beloved (Israel who was to have been His faithful wife) in My house when she has done many vile deeds? Can the sacrificial flesh take away from you your disaster, so that you can rejoice? (God was not interested in their sacrifices but in their broken and contrite hearts)" 16 The LORD called your name, "A green olive tree, beautiful in fruit and form"; With the noise of a great tumult He has kindled fire on it, and its branches are worthless (Paul has explained in Romans 11 these were broken off). 17 And the LORD of hosts, who planted you, has pronounced evil against you because of the evil of the house of Israel and of the house of Judah, which they have done to provoke Me by offering up sacrifices to Baal.  (Jeremiah 11:15-17)

Jesus over 600 years later warned His Israel that...

"the kingdom of God will be taken away from you, and be given to a nation producing the fruit of it. (Matthew 21:43).

W E Vine comments that...

the Gentile who, though now a believer, is, naturally, a member of those nations which God had suffered to walk in their own ways (Ac 14:16), and who therefore remained unfaithful to God. The wild olive being a tree of comparatively little value, this part of the metaphor sets in contrast the glorious position of relationship into which God had brought Israel on the ground of covenant promise. The process of grafting is almost invariably that of putting the good shoot into the inferior stock. Whenever the reverse process was adopted it was to invigorate the fruitful stock and not to fertilize the wild shoot. What is set forth here, therefore, does not correspond actually to either method, but is selected in order to suit the subject, namely, the enrichment of individual Gentiles by their admission to the blessings forfeited by some of God’s ancient people through their unbelief.

AND BECAME PARTAKER WITH THEM OF THE RICH ROOT OF THE OLIVE TREE: kai sugkoinonos tes rhizes tes piotetos tes elaias egenou (2SAMI):

Partaker (4791) (sugkoinonos from sun = with + koinonós = companion, partner) means co-participant or partaker together with others.

The New Living Translation phrases this verse this way...

But some of these branches from Abraham's tree, some of the Jews, have been broken off. And you Gentiles, who were branches from a wild olive tree, were grafted in. So now you also receive the blessing God has promised Abraham and his children, sharing in God's rich nourishment of his special olive tree.

The discerning reader needs to be careful when reading some of the commentaries on this section of Romans. For example the respected devotional commentary by Matthew Henry has some very confusing statements on Romans 11:17. Henry writes that...

"those that God grafts into the church he finds wild and barren, and good for nothing. Men graft to mend the tree; but God grafts to mend the branch. The church of God is an olive-tree,  flourishing and fruitful as an olive (Ps. 52:8; Hos. 14:6)...The Gentiles, being grafted into the church, partake of the same privileges that the Jews did, the root and fatness. The olive-tree is the visible church (Henry, M. Matthew Henry's Commentary on the Whole Bible: Complete and Unabridged)

Clearly Matthew Henry is misinterpreting the rich root of the olive tree as the church. Notice also that Henry goes so far as to find the "church" in Old Testament passages like Hosea 14:6 which are clearly (in context) speaking of the nation of Israel. In a sense Matthew Henry is doing the very thing Paul warns Gentile believers against in this section of Romans.

Warren Wiersbe agrees and cautions that 

To say that the olive tree, with its natural and grafted branches, is a picture of the church would be a great mistake. (Wiersbe, W. W. The Bible Exposition Commentary.  Wheaton, Ill.: Victor Books) (Bolding added)

It should also be noted that the fact that true believers living today are spiritual children of Abraham does not mean that the church is "spiritual Israel." Yes as Paul said "if you belong to Christ, then you are Abraham's offspring, heirs according to promise." (Gal 3:29)  Gentile believers who are not physical children of Abraham are his spiritual children in the sense that they have followed the pattern of his faith. All believers, Jew or Gentile, in fact are heirs of the spiritual blessings of the Abrahamic Covenant. The Bible Knowledge Commentary (on Galatians 3:29) however cautions that...

Any discussion of the seed (offspring) of Abraham must first take into account his natural seed (offspring), the descendants of Jacob in the 12 tribes. Within this natural seed there is a believing remnant of Jews who will one day inherit the Abrahamic promises directed specifically to them (cf. Ro 9:6, 8).

But there is also the spiritual seed of Abraham who are not Jews. These are the Gentiles who believe and become Abraham’s spiritual seed. They inherit the promise of justification by faith as Paul explained... (cf. Gal. 3:6-9). To suggest, as amillenarians do (Ed note: and as many of the older commentators like Matthew Henry), that Gentile believers inherit the national promises given to the believing Jewish remnant (Ed note: e.g., the specific promises of the land in Genesis 15:18) promises which God has not yet literally fulfilled but which He will fulfill in the Millennium) —that the church thus supplants Israel or is the “new Israel”—is to read into these verses what is not there. (Walvoord, J. F., Zuck, R. B., & Dallas Theological Seminary. The Bible Knowledge Commentary. Wheaton, IL: Victor Books) (Bolding added)  (See also discussion of the phrase Israel of God)

Arrogant (2620) (katakauchaomai from katá = against + kaucháomai = boast over a privilege or possession. The root is auchen  = neck which vain persons are apt to carry in proud manner) means to exult or boast against or to boast arrogantly. In Ro 11:18 the Gentile believers are contrasted with the "branches" those Jews who rejected Christ. The verb is in the with a negative which can be more accurately translated as "stop being arrogant toward" the unbelieving Jews. The point is that this aberrant attitude had already begun to creep into the Roman church which by all of Paul's testimony was a good church. How easily this leaven of subtle anti-Semitism can begin to permeate even the best of churches today! We must diligently guard against this attitude toward the Jews or the nation of Israel.

How could a Gentile, a wild olive, who had been grafting in (by faith) demonstrate his or her arrogance? One way is by saying that the Jews who did not believe should have believed! Another way is to assume a  holier-than-thou attitude toward the Jews.

It is tragic and lamentable that, throughout much of church history, Jewish converts to Christ have often been subjected to attitudes of Gentile superiority and been shunned or reluctantly accepted into Christian fellowship. Paul anticipated that, in spite of this clear truth, some of his Gentile readers would continue to argue against him (see Romans 11:19).

As an aside I would pose this question: What happened in this regard to the Early Church Fathers? Why did not many of them see this truth? Even Martin Luther in his latter years wrote a vicious pamphlet vilifying the Jews because they would not accept Messiah! (See a sample link from a non-believing Jewish source) If Martin Luther could fall into such deception, let every man take heed lest he fall!

William Newell comments...

How few of us Gentile believers understand and bear in mind that we are beneficiaries of those promises which God lodged in Abraham as a root of promise, --all the promises we inherit in Christ! (Romans 11).

BUT IF YOU ARE ARROGANT REMEMBER THAT IT IS NOT YOU WHO SUPPORTS THE ROOT BUT THE ROOT SUPPORTS YOU: ei de katakauchasai (2SPMI) ou su ten rhizan bastazeis (2SPAI) alla e rhiza se: (4:16; Zechariah 8:20-23; John 10:16; Galatians 3:29; Ephesians 2:19,20)

Root = spiritual promises and blessings that flow through the Abrahamic Covenant. In a sense even the New Covenant was also actually given to Israel first. E.g. Jeremiah writes...

"Behold, days are coming," declares the LORD, "when I will make a new covenant with the house of Israel and with the house of Judah, (with who? is it with the Gentiles?) 32 not like the covenant which I made with their fathers in the day I took them by the hand to bring them out of the land of Egypt, My covenant which they broke, although I was a husband to them," declares the LORD. (for more on this subject see the New Covenant in the Old Testament) (Jeremiah 31:31-32)

And who did Jesus inaugurate the New Covenant with? Were there any Gentiles present? And who did the Spirit Who was promised as part of the blessing of the New Covenant first come upon in Acts 2? And who composed the predominant makeup of disciples in the early church (probably until about 100AD)? (Answer: Jewish believers). Clearly we Gentiles are "Johnny come lately" and thus have no grounds whatsoever for spiritual pride in regard to our salvation.

There is never any ground for a believer from among the Gentiles to hold a Jew as such in any measure of contempt or inferiority. Paul warned the Gentiles that they were obligated to Israel, and therefore they dared not boast of their new spiritual position (Ro 11:18-21). The Gentiles entered into God’s plan because of faith, and not because of anything good they had done. Paul was discussing the Gentiles collectively, and not the individual experience of one believer or another.

No matter how far Israel may stray from the truth of God, the roots (the covenant promises to Abraham, Isaac and Jacob were made by a covenant keeping God!) are still good. God is still the “God of Abraham, and the God of Isaac, and the God of Jacob” (Ex 3:6 Mt 22:32). He will keep His promises to the patriarchs and to any and all natural Jews who enter in by grace through faith.

Praise God that His the keeping of His covenants does not depend on us but is rooted in His attributes such as Faithfulness, Love, Mercy, Immutable, Good, Eternal