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### Synthetic Division

```Date: 05/06/2003 at 22:23:17
From: Noel
Subject: Algebra

(x^2-6x+9)/(x-3)
```

```
Date: 05/07/2003 at 03:38:15
From: Doctor Luis
Subject: Re: Algebra

Hi Noel,

Here's how you divide a polynomial by (x-a):

First draw a grid like this, showing the powers of x and with the
coefficients below.

x^2     x    1
-------------------
3 |   +1     -6    +9
--|-------------------
|

If a power of x is missing, make sure you put a 0 as the coefficient.
On the side, put the negative of the number you're dividing by. If
you're dividing by x-3, put -(-3)=+3. If you're dividing by x+1 put -1
there.

Done drawing it? Good.

First, bring down the 1. (the first number inside the box)

x^2     x    1
-------------------
3 |   +1     -6    +9
--|-------------------
|    1

Next, multiply the number you brought down by the number on the side.
Since I brought down a 1, I multiply 3*1 = 3. Add this number to the
second number (here it's -6), and write that number down (3-6 = -3) in
the next spot.

x^2     x    1
-------------------
3 |   +1     -6    +9
--|-------------------
|    1     -3

Now repeat this process. Multiply the 3 times -3 and add it to the 9.
3*(-3)+9 = 0.

x^2     x    1
-------------------
3 |   +1     -6    +9
--|-------------------
|    1     -3     0

This last number should be a zero (for our problem it is), if the
polynomial x-3 divides x^2-6x+9. If it isn't, then it represents a
remainder.

x^2     x    1
-------------------
3 |   +1     -6    +9
--|--------------|----
|    1     -3  |  0
|--------------|----
x      1

You start counting one power of x less on the bottom row. Since we
started with x^2, the next lower power is x. You can see the answer
is x-3. Remember that the last number is actually a remainder
(technically, it represents the -1 power).

Therefore,

x^2 - 6x + 9
------------- = x - 3
x - 3

This process is called synthetic division.

Let's try a more difficult example that'll teach you more about
synthetic division.

How about dividing x^3 + 3x + 8 by x+1 ? I won't draw the powers of x
this time, because you already know what the numbers in the box mean.
You can imagine them to be there in your mind. Note that the 0 is
there because there's no x^2 power.

|
-1 | 1   0   3   8
---|-----------|----
| 1  -1   4 | 4

See how that worked? Bring down the 1. Multiply by -1 and add to 0,
and you get -1. Multiply -1 by -1 and add to 3, you get 4. Multiply 4
by -1 and add to 8, and you get 4. This time, we did get a remainder.

Reading the table back, we see that the solution to our problem is

x^3 + 3x + 8                     4
-------------- = x^2 - x + 4 + --------
x + 1                       x + 1

See what happens when you get a remainder term? There's an extra
4/(x+1) term at the end. That's how you write the remainder. The thing
at the bottom of the remainder is just the thing you're dividing by.

I hope this helped! Let us know if you have any other questions.

- Doctor Luis, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Polynomials

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