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### Synthetic Division

```
Date: 12/22/96 at 23:58:18
From: David P Mcclain
Subject: Synthetic division

college student and still don't understand it.
```

```
Date: 12/24/96 at 15:02:24
From: Doctor Rob
Subject: Re: Synthetic division

Synthetic division is just a short-hand way of dividing a polynomial
by a linear monic polynomial.  In other words, one divides, using long
division of polynomials, say:

c3*x^2 + (c3*r+c2)*x + (c3*r^2+c2*r+c1)
--------------------------------------------------
x - r ) c3*x^3 + c2*x^2        + c1*x               + c0
c3*x^3 - c3*r*x^2
-----------------
(c3*r+c2)x^2        + c1*x
(c3*r+c2)*x^2 - (c3*r^2+c2*r)*x
-------------------------------
(c3*r^2+c2*r+c1)*x               + c0
(c3*r^2+c2*r+c1)*x - (c3*r^3+c2*r^2+c1*r)
-----------------------------------------
(c3*r^3+c2*r^2+c1*r+c0)

The idea is to remove all the redundancy from this tableau.

1. By keeping columns of coefficients, we can eliminate all the powers
of x, and the + signs between terms.

c3       (c3*r+c2)       (c3*r^2+c2*r+c1)
--------------------------------------------------
1  -r ) c3     c2              c1                     c0
c3  -c3*r
---------
c3*r+c2              c1
c3*r+c2  -(c3*r^2+c2*r)
-----------------------
c3*r^2+c2*r+c1                   + c0
c3*r^2+c2*r+c1  -(c3*r^3+c2*r^2+c1*r)
-------------------------------------
c3*r^3+c2*r^2+c1*r+c0

2. The leading terms of the subtracted polynomials are always equal to
the ones immediately above, so they are redundant and can be
deleted.

3. The terms "brought down" are redundant and can thus be eliminated:

c3       (c3*r+c2)       (c3*r^2+c2*r+c1)
--------------------------------------------------
1  -r ) c3     c2              c1                     c0
-c3*r
---------
c3*r+c2
-(c3*r^2+c2*r)
-----------------------
c3*r^2+c2*r+c1
-(c3*r^3+c2*r^2+c1*r)
-------------------------------------
c3*r^3+c2*r^2+c1*r+c0

dropping all the - signs:

c3       (c3*r+c2)       (c3*r^2+c2*r+c1)
--------------------------------------------------
1   r ) c3     c2              c1                     c0
c3*r
----
c3*r+c2
c3*r^2+c2*r
--------------
c3*r^2+c2*r+c1
c3*r^3+c2*r^2+c1*r
---------------------
c3*r^3+c2*r^2+c1*r+c0

5. The quotient above the line is the same as the quantities below the
line (if we ignore the last one, which is the remainder), so it can
be dropped.  The horizontal line over the dividend can be removed,
and all the other lines brought together to make a single
horizontal line down below.  The 1 is redundant, too, since it is
always there, so it can be understood.

r ) c3       c2              c1                      c0
---         c3*r     c3*r^2+c2*r      c3*r^3+c2*r^2+c1*r
---------------------------------------------------
c3  c3*r+c2  c3*r^2+c2*r+c1 | c3*r^3+c2*r^2+c1*r+c0
-----------------------
At this point, the mechanics of synthetic division can be readily
seen:

1. Write r down, and put a line to the right and below it.  This
represents the divisor.

2. Further to the right, list the coefficients of the dividend
polynomial.  Don't forget to put in zeros if powers of the variable
x are skipped.

3. Leave a blank line, then draw a horizontal line for the additions
to follow.

4. Bring down the first coefficient below the line.

5. Multiply that last number by the number r, and put it above the
line and in the next column to the right.

6. Add the two numbers in this column, and put the sum in the same
column below the line.

7. If we are not out of columns, go back to step 5.

8. Draw a line to the left and below the last number in the last line.

9. The quotient is read off from the last line, coefficient by
coefficient, left to right, ending at the vertical line.  The
remainder is the last number in the last line, to the right of the
vertical line.

Example:  Divide (x - 3) into x^4 + x^3 - 7*x + 11

3 ) 1   1   0  -7  11
--      3  12  36  87
-------------------
1   4  12  29 | 98
----

So the quotient is x^3 + 4*x^2 + 12*x + 29, and the remainder is 98.

If the divisor is x + 7, then put r = -7, for instance.  If the
divisor is of the form 3*x - 14, then instead divide by x - 14/3, and
then divide the quotient and remainder by 3, coefficient by
coefficient.  This obviously all works whatever the name of the
variable (x in this message).

Something more complicated must be used if the divisor has degree
higher than 1.

I hope that this is clear.  If not, please write back for more help.

-Doctor Rob,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Polynomials

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