You do * polynomial division* the way you do long
division of numbers. It's difficult to describe the general procedure
in words, so I'll work through some examples step-by-step.

* Example.* Find the quotient and remainder
when is divided by .

You may also see this kind of problem written like this: "Perform the division ."

Set up the division the way you'd set up division of numbers. To start, look at the first term (x) in and the first term ( ) in .

Ask yourself: "What times x gives ?"

You can see that x works, so put it on top:

Next, multiply the by the x on top, and put the result under . Line up terms with the same power of x:

Subtract: .

(When you subtract, *be careful of the signs!* In this case,
the terms cancel, but .)

Next, bring down the -3 and put it next to the :

Look at the first term (x) in and the first term ( ) in .

Ask yourself: "What times x gives ?"

You can see that 4 works, so put it on top:

Multiply the by the 4 on top, and put the result under . Line up terms with the same power of x:

(You don't multiply by the on top, just the 4; you already multiplied by the "x" in in an earlier step.)

Subtract: .

At this point, the "x" of " " doesn't go into 5, so the division is finished.

The quotient is , the expression on the top. And the remainder is 5.

If you're just asked for the quotient and remainder, you're done.

If the problem asked you to do the division , then you'd write it this way:

The quotient goes in front. The remainder 5 goes on top of the fraction. The expression , which was on the bottom of the original fraction, goes on the bottom of the new fraction.

If you have trouble remembering where everything goes, think about
how you convert * improper fractions* to * mixed numbers*. Let's say you want to convert to a mixed number. To do this, you divide
38 by 7:

Then .

However, if you think about it, --- which you
read as "5 * and* " --- means " ", so

This is the same as what I did with the polynomials: The quotient 5 goes in front. The remainder 3 goes on top of the fraction. The expression 7, which was on the bottom of the original fraction , goes on the bottom of the new fraction.

In other words, as often happens when you're doing algebra, you can figure out "what to do with variables" by thinking about "what you know to do with numbers".

* Example.* Find the quotient and remainder
when is divided by .

Since is missing an term, I'll write " " as a placeholder. It is optional, but this makes it less likely that you'll make a mistake when you do the subtraction.

Look at the first term (x) in and the first term ( ) in :

Ask yourself: "What times x gives ?"

You can see that works, so put it on top:

Next, multiply by the on top, and put the result under . Line up terms with the same power of x:

Note that the goes under the I put in as a placeholder.

Subtract: .

Next, bring down the and put it next to the :

Look at the first term (x) in and the first term ( ) in :

Ask yourself: "What times x gives ?"

You can see that works, so put it on top:

Multiply by the on top (just the " " --- you multiplied by the " "), and put the result under :

Subtract: .

Bring down the 1 --- since , I just write the 1:

Since does not go into 1, the division is finished:

The quotient is and the remainder is 1.

In fraction form, this would be written as

* Example.* Find the quotient and remainder
when is divided by .

Look at the first term ( ) in and the first term ( ) in :

Ask yourself: "What times gives ?"

You can see that works, so put it on top:

Next, multiply by the on top, and put the result under . Line up terms with the same power of x:

Subtract: .

Next, bring down the 12 and put it next to the :

Look at the first term ( ) in and the first term ( ) in :

Ask yourself: "What times gives ?"

You can see that 5 works, so put it on top:

Multiply by the 5 on top (just the "5" --- you already multiplied by the " "), and put the result under :

Subtract: .

Since does not go into 7, the division is finished.

The quotient is and the remainder is 7. In fraction form, this would be written as

* Example.*

* Example.*

Sine the remainder is 0, this shows that *factors* as .

* The Remainder Theorem.* If is a polynomial, the remainder when is divided by is .

* Example.* Divide by :

The remainder is 47. On the other hand,

The remainder is the same as .

* The Root Theorem.* divides evenly if and only if
(i.e. is a root of ).

* Example.* Let .

Therefore, should divide . In fact,

* Example.* Let . Then

Since and are factors, and should be roots --- and they are:

* Example.* If you know or can guess a factor,
you can sometimes complete a factorization by long division. For
example has as a root. By the fact I stated earlier, this means
that is a factor. Divide into to get .

Therefore,

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Copyright 2015 by Bruce Ikenaga