Please analyze Polonius in Act One, scene three of Shakespeare's Hamlet?

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In Shakespeare's Hamlet, Polonius is a courtier to newly crowned Claudius.

Polonius, Laertes's and Ophelia's father, is an elderly and long-winded courtier and chief counselor in the Danish court. Polonius demonstrates a propensity for hypocrisy...

Anxious (we can infer) to make himself indispensible to his new King, Polonius places himself in situations whereby he attempts to prove he is knowledgeable. In reality, Polonius is an idiot. In Act One, scene three, he is a character Shakespeare uses to deliver excellent advice to his children (in particular, Laertes), but he is unable to follow the advice himself. Some of his lines are still quoted today. The first quote deals simply with good manners.

Be thou familiar, but by no means vulgar. (65)

The next quote counsels that one be a good listener, but keep his thoughts to himself. (This is impossible for Polonious.) The King and Queen hear everything he thinks, especially things that have no bearing on the topic at hand.

Give every man thy ear, but few thy voice... (72)

The following line we have come to recognize as clothes make the man:

...the apparel oft proclaims the man... (76)

A very famous line that sounds like something from the Bible or Benjamin Franklin is...

Neither a borrower nor a lender be... (79)

Perhaps the best words are those that speak directly to the core of the plot: honesty as opposed to duplicity

This above all: to thine own self be true,

And it must follow, as the night the day,

Thou canst not then be false to any man. (82-84)

All of this information is excellent, but Polonius is better at sharing it with others than following it himself. (We will find him to be sneaky and manipulative: this makes him a hypocrite.)  

When Laertes leaves, Polonius wants to know of his advice to Ophelia. She reports her brother's words of caution about Hamlet. Polonius asks his daughter what she thinks of Hamlet's recent "offers of affection." She is cautious, not sure what to think. Polonius decides for her: he insists Hamlet is lying and tells Ophelia that she really knows nothing—she is like a baby. She should consider Hamlet's offers lies...insisting they are not sincere (though Polonius has nothing to base this on), and demands that she keep her distance so that she doesn't get pregnant, making him look foolish.

Marry, I'll teach you. Think yourself a baby,

That you have ta'en these tenders for true pay,

Which are not sterling. Tender yourself more dearly,

Or—not to crack the wind of the poor phrase,

Running it thus—you'll tender me a fool. (111-115)

Polonius does not give his daughter credit for being at all intelligent, and shows no real affection for her. He is more interested in his image. He orders her to stay away from Hamlet.

While Polonius tries to pretend that he knows all—about life and about love—we must question what he says. He is unable to be the honorable and wise man he wants Laertes to be (as we compare the advice he gives to the way he acts). He speaks of knowing about a young man in love, but we can hardly imagine Polonius a man of this sort, for he seems to care little for anyone but himself.

I do know,

When the blood burns, how prodigal the soul

Lends the tongue vows. (122-124)

Polonius is a many of many words—empty words. He is egotistical and arrogant. He is not an affectionate, caring father to Ophelia. He gives us little to respect.