1. Dividing
Polynomials

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

3. 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

4. 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

5. 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

6. 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

7. 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)

8. Using
Synthetic Division

9. 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.

10. 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

11. 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

12. 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).

13. 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.

14. 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

15. 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!

16. 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.

17. 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.

18. 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)