1. Dividing
Polynomials

2. Rational Expressions
A rational expression is a mathematical expression
containing polynomials in the numerator and denominator.
In all rational expressions, the denominator can not be
equal to zero, since division by zero is undefined.
Therefore, values of the variable that would make the
denominator equal to zero must be listed. These values
are called Restricted Values.They should be listed for
every rational expression.

3. Listing Restricted Values
x ≠ 0
x ≠ -3
x ≠ 0, -3
x ≠ 0, ±3
x ≠ 4, - 5
x ≠ ±4
xy ≠ 0
(3a - b) ≠ 0
(3a + b) ≠ 0
x ≠ 0
(x - y) ≠ 0
(x + y) ≠ 0

4. Division by a Monomial
= 5xy
x ≠ 0
y ≠ 0
= - 9a2 b
a ≠ 0
b ≠ 0
= x2 - 2x + 3
x ≠ 0

5. Basic Long Division - A Review
Quotient 5 7 8 8
4 0 6 3
Dividend
Divisor
5 6 7 1
6 4 7
Remainder
4631 = 8 x
Dividend = Divisor • Quotient + Remainder

6. Division of a Polynomial by a Binomial
Dividend = Divisor x Quotient + Remainder
P(x) = D(x) Q(x) + R(x)
This is called the Division Statement or Division Algorithm.
Divide: x2 + 7x + 2 ÷ (x + 2)
1. The polynomial
must be in descending
order of powers. Any
missing terms are to be
filled with a zero
placeholder.
Multiply x
+ 5
x + 2
x2 + 7x + 2
x2 + 2x
x(x + 2)
= x2 + 2x 5x
+ 2
2. Only the first term is
used when doing the
division.
5x + 10
Divide x2 x -8
= x
3. Multiply your answer
with the entire divisor.
4. Subtract, bring down
the next term and repeat
the process.
P(x) =
(x + 2)
(x + 5)
- 8

7. Division by a Binomial 2m2 + m - 4 2m + 1 4m3 + 4m2 - 7m - 3 4m3 + 2m2
Multiply
Multiply
Multiply
2m2
+ m
- 4
2m + 1
4m3 + 4m2 - 7m - 3
4m3 + 2m2
2m2
- 7m
2m2 + m
- 8m
- 3
- 8m - 4 1
P(x) = D(x) Q(x) + R(x)
P(m) = (2m + 1)(2m2 + m - 4) + 1

8. Division by a Binomial
4x2
- 3x
+ 2
x - 2
4x3 - 11x2 + 8x + 10
4x x2
- 3x2
+ 8x
- 3x2 + 6x 2x
+ 10 2x 14
P(x) = D(x) Q(x) + R(x)
P(x) = (x - 2)(4x2 - 3x + 2) + 14

9. Division by a Binomial
Divide: (8m3 - 1) ÷ (2m - 1)
4m2
+ 2m
+ 1
2m - 1
8m3 + 0m2 + 0m - 1
8m3 - 4m2
4m2
+ 0m
4m2 - 2m 2m
- 1
2m - 1
P(m) = (2m - 1)(4m2 + 2m + 1)

10. Using
Synthetic Division

11. Quotient Divisor Dividend Remainder + =
The Division Statement
Quotient
Divisor
Dividend
Remainder
= (7) X (9) +
Dividend =
Divisor X
Quotient
Remainder
For any division problem:
Dividend = (Divisor)(Quotient) + Remainder
P(x) = D(x) • Q(x) + R(x)

12. 5 1 -2 -33 90 5 15 -90 3 -18 Quotient Rem Synthetic Division 1
Divide x3 - 2x x by (x - 5) using synthetic division. 5 1 -2
-33 90
1. Write only the constant
term of the divisor, and
the coefficients of the
dividend.
Add
Add
Add 5 15
Multiply
Multiply
-90
Multiply 1 3
-18
2. Bring down the first term
of the dividend.
Quotient
Rem
3. Multiply 1 by 5, record
the product and add.
Written as x2 + 3x - 18
4. Multiply 3 by 5, record
the product and add.
Using the division statement:
P(x) = (x - 5)(x2 + 3x - 18)
5. Multiply -18 by 5, record
the product and add.

13. Using Synthetic Division
Divide: (x4 - 2x3 + x2 + 12x - 6) ÷ (x - 2) 2 2 2 28 1 1 14 22
P(x) = D(x) Q(x) + R(x)
What does this
mean ????
P(x) =
(x - 2)
(x3 + x + 14)
+ 22
Remainder
Divisor
Quotient

14. Using Synthetic Division
1. (4x3 - 11x2 + 8x + 6) ÷ (x - 2)
P(x) = (x - 2)(4x2 - 3x + 2) + 10 8 -6 4 4 3 2 10
2. (2x3 - 2x2 + 3x + 3) ÷ (x - 1)
P(x) = (x - 1)(2x2 + 3) + 6

15. The Remainder Theorem Given f(x) = x3 - 4x2 + 5x + 1, determine
the remainder when f(x) is divided by x - 1. 1
The remainder is 3. 1 -3 2 1 -3 2 3
NOTE: f(1) gives
the same answer
as the remainder
using synthetic
division.
Using f(x) = x3 - 4x2 + 5x + 1, find f(1):
f(1) = (1)3 - 4(1)2 + 5(1) + 1 =
= 3
Therefore f(1) is equal to the remainder. In other words,
when the polynomial x3 - 4x2 + 5x + 1 is divided by x - 1,
the remainder is f(1).

16. The Remainder Theorem:
When a polynomial f(x) is divided by x - b, the
remainder is f(b). [Think x - b, then x = b.]
Find the remainder when x3 - 4x2 + 5x - 1 is divided by:
a) x - 2
b) x + 1
Find f(2):
f(2) = (2)3 - 4(2)2 + 5(2) - 1 =
= 1
Find f(-1):
f(-1) = (-1)3 - 4(-1)2 + 5(-1) - 1 =
= -11
The remainder is 1.
The remainder is -11.

17. Divide: (6x3 - 7x2 + 10x + 6) ÷ (2x - 1) 1 2 6 -7 10 6
Dividing by ax - b
Divide: (6x3 - 7x2 + 10x + 6) ÷ (2x - 1) 1 2
To use synthetic division,
the coefficient of x in the
divisor must be 1. Therefore,
you factor the divisor: 3 -2 4 6 -4 8 10
P(x) = (x - 1/2)(6x2 - 4x + 8) + 10
Note that the
divisor is
P(x) = (x - 1/2)(2)(3x2 - 2x + 4) + 10
Note that the
quotient has a
factor of 2 in it.
Multiply the factor
with the divisor.
P(x) = (2x - 1)(3x2 - 2x + 4) + 10

18. In general I find that it is easier to do
Dividing by ax - b
Divide: (6a3 + 4a2 + 9a + 6) ÷ (3a + 2)
The divisor is -2 3 -4 -6 6 9
P(x) = (a + 2/3)(6a2 + 9)
P(x) = (a + 2/3)(3)(2a2 + 3)
Factor out the 3 from
the quotient.
P(x) = 3(a + 2/3)(2a2 + 3)
Multiply the 3 through
the divisor.
P(x) = (3a + 2)(2a2 + 3)
In general I find that it is easier to do
problems like the last two using long division!

19. Finding the Remainder When Dividing by ax - b
Find the remainder when P(x) = 2x3 + x2 + 5x - 1
is divided by 2x - 1.
For the divisor 2x - 1,
factor out the a to
determine the value of b.
P(1/2) = 2(1/2)3 + (1/2)2 + 5(1/2) - 1
= 2(1/8) +(1/4) + 5/2 - 1
= 1/4 + 1/4 + 5/2 - 1 = 4 1
+ 1
+ 10
- 4
= 2
The remainder is 2.

20. Factor Theorem If f(x) is a polynomial function, then
(x – c) is a factor of f(x) if and only if f(c) = 0.
Ex: Given f(x) = x2 + 3x – 18, (x + 6) is a factor of f(x) if and only if f(-6) = 0.
(-6)2 + 3(-6) – 18 = 0
Ex: Given f(x) = x2 + 3x – 18, (x - 3) is a factor of f(x) if and only if f(3) = 0.
(3)2 + 3(3) – 18 = 0
We can check these results by using synthetic division.

21. You can also use synthetic division to find factors of a polynomial...
Example: Given that (x + 2) is a factor of f(x), factor the polynomial f(x) = x3 – 13x2 + 24x + 108
Since (x + 2) is a factor, –2 is a zero of the function…
We can use synthetic division to find the other factors...
– – –2 30
–108 1
–15 54
This means that you can write
x3 – 13x2 + 24x = (x + 2)(x2 – 15x + 54)
This is called the depressed polynomial
Factor this = (x + 2)(x – 9)(x – 6)
The complete factorization is (x + 2)(x – 9)(x – 6)