This document discusses different methods for dividing polynomials: 1. Simple division - dividing a polynomial by a monomial by distributing the monomial factor. 2. Long division - the standard long division algorithm for dividing polynomials, where you repeatedly subtract multiples of the divisor polynomial from the dividend polynomial. 3. Synthetic division - a compact algorithm for dividing a polynomial by a polynomial where the leading coefficient of the divisor is 1, by writing the terms in descending order and performing operations on the coefficients.

1. Dividing
Polynomials

2. Simple Division -
dividing a polynomial by a monomial
2 2 2 2
6 3 9
3
r s rs r s
1.
rs
+ −
=
6r2
s2
3rs
+
3rs2
3rs
−
9r2
s
3rs
= 2rs + s − 3r

3. Simplify
=
3a2
b
3ab
+
6a3
b2
3ab
+
18ab
3ab
= a + 2a2
b + 6
2 3 2
3 6 18
3
a b a b ab
2.
ab
+ +

4. Simplify
=
12x2
y
3x
−
3x
3x
= 4xy −1
2
12 3
3
x y x
3.
x
−

5. Long Division -
divide a polynomial by a polynomial
•Think back to long division from 3rd grade.
•How many times does the divisor go into the
dividend? Put that number on top.
•Multiply that number by the divisor and put the
result under the dividend.
•Subtract and bring down the next number in the
dividend. Repeat until you have used all the
numbers in the dividend.

6. -( )
x + 3 x2
− 5x − 24)
x
x2 + 3x
- 8x
- 8
- 24
- 8x - 24
0
-( )
x2
/x = x
2
5 24
3
x x
4.
x
− −
+
-8x/x = -8

7. -( )
-( )
-( )
h − 4 h3
+ 0h2
−11h + 28)
h2
h3 - 4h2
4h2
+ 4h
- 11h
4h2
- 16h
5h
h3
/h = h2
( ) ( )
13
11 28 45. h h h
−
− + −
+ 284h2
/h = 4h
+ 5
5h - 20
48
+
48
h − 4
5h/h = 5

8. Synthetic Division -
( )4 2
6 : 5 4 6 ( 3)Ex x x x x− + + ÷ −
To use synthetic division:
•There must be a coefficient for
every possible power of the variable.
•The divisor must have a leading
coefficient of 1.
divide a polynomial by a polynomial

9. Step #1: Write the terms of the
polynomial so the degrees are in
descending order.
5x4
+ 0x3
− 4x2
+ x + 6
Since the numerator does not
contain all the powers of x,
you must include a 0 for the x
3
.
( )4 2
5 4 6 ( 3)x x x x− + + ÷ −

10. Step #1: Write the terms of the
polynomial so the degrees are in
descending order.
5x4
+ 0x3
− 4x2
+ x + 6
Since the numerator does not
contain all the powers of x,
you must include a 0 for the x
3
.
( )4 2
5 4 6 ( 3)x x x x− + + ÷ −