Synthetic Division

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

Please teach me about synthetic division!  I am an over-the-hill 
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

4. Instead of subtracting negatives, we can add positives.  This means
   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/