
Review of Long Division
Example
Use long division to calculate
495/12
and will write the steps for this process without using any numbers.
Solution



4 
1 
3/12 
12 
 
4 
9 
5 



4 
8 





1 
5 




1 
2 





3 

We see that we follow the steps:

Write it in long division form.

Determine what we need to multiply the quotient by to get the first
term.

Place that number on top of the long division sign.

Multiply that number by the quotient and place the product below.

Subtract

Repeat the process until the degree of the difference is smaller
than the degree of the quotient.

Write as sum of the top numbers + remainder/quotient.
P(x)/D(x) = Q(x) +
R(x)/D(x)
Below is a nonsintactical version of a computer program:
while (degree of denominator < degree of remainder)
do
{
divide first term of remainder by first term of denominator and place
above
quotient line;
multiply result by denominator and place product under the remainder;
subtract product from remainder for new remainder;
}
Write expression above the quotient line + remainder/denominator;
Exercises

(3x^{2} + 5x + 7)/(x + 1)

(2x^{4} + x  1)/(x^{2} + 3x + 1)

Synthetic Division
For the special case that the denominator is of the form x 
r, we can use
a shorthand version of polynomial division called synthetic division. Here
is a step by step method for synthetic division for P(x)/(x 
r):
Step 1:
Drop all the x's filling in zeros where appropriate and set up the division
r  a b c d
and place a horizontal line leaving space between the numbers
and the line.
Step 2:
Put the first coefficient under the line.
r 
a b c d
a
Step 3:
Multiply r by the number under the line and place the product below the second
coefficient.
r 
a b c d
ra
a
Step 4:
Add the second column and place the sum below the line.
Step 5:
Repeat steps 3 and 4 until there are no more columns.
Step 6:
The last number is the remainder and the first numbers are the coefficients
of the polynomial Q(x)
Example:
Use synthetic division to find
(2x^{3} + x + 7)/(x + 1)

_ 
_ 
_ 
_ 
_ 
1 
 
2 
0 
1 
7 



2 
2 
1 

_ 
_ 
_ 
_ 
_ 


2 
2 
1 
8 






Solution:
2x^{2} + 2x + 1 + 8/(x + 1)
Steps:

Bring down the 2.

Multiply (1)(2) = 2 and place it under the 0.

Add 0 + 2 = 2 and place it in the third row.

Multiply (1)(2) = 2 and place it under the 1.

Add 1 + (2) = 1 and place it in the third row.

Multiply (1)(1) = 1 and place it under the 7.

Add 7 + 1 = 8 and place it in the third row.

Write 2x^{2} + 2x + 1 + 8/(x + 1)

The Remainder Theorem
The Remainder Theorem For any polynomial P(x)
P(r) = the remainder of
P(x)/(x  r)
in particular, if P(r) = 0 then the
remainder is also 0. 
Proof:
P(x)/(x  r) =
Q(x) + R/(x  r)
Multiply both sides by x  r to get
P(x) = Q(x)(x  r) + R
Plugging in r, we have
P(r) = Q(r)(r  r) + R = R.
Exercise:
Verify that 2 is a root of
x^{3} 
3x^{2} + x + 2
using the remainder theorem.