The document discusses polynomial division algorithms. It introduces long division and synthetic division as methods for dividing polynomials. Long division is analogous to dividing numbers, while synthetic division is simpler but only applies when dividing a polynomial by a monomial. The key points are: - Long division allows dividing any polynomial P(x) by any polynomial D(x) to obtain a quotient Q(x) and remainder R(x) such that P(x) = Q(x)D(x) + R(x) and the degree of R(x) is less than the degree of D(x). - Synthetic division is more efficient than long division when dividing a polynomial by a monomial of the form (

1. Methods of Division

2. Methods of Division
The questions of factorability and finding roots of
real polynomials (polynomials with real coefficients)
are the main themes of this chapter.

3. Methods of Division
We start with two division algorithms for polynomials
of a single variable x; long division and synthetic
division.
The questions of factorability and finding roots of
real polynomials (polynomials with real coefficients)
are the main themes of this chapter.

4. Methods of Division
We start with two division algorithms for polynomials
of a single variable x; long division and synthetic
division.
The questions of factorability and finding roots of
real polynomials (polynomials with real coefficients)
are the main themes of this chapter.
Long division of polynomials is analogous to the
long division of numbers and it’s a general division
method for P(x) ÷ D(x).

5. Methods of Division
We start with two division algorithms for polynomials
of a single variable x; long division and synthetic
division.
The questions of factorability and finding roots of
real polynomials (polynomials with real coefficients)
are the main themes of this chapter.
Long division of polynomials is analogous to the
long division of numbers and it’s a general division
method for P(x) ÷ D(x).
Synthetic division is a simpler method for dividing
polynomials P(x) by monomials of the form (x – c),
i.e. P(x) ÷ (x – c).

6. Methods of Division
We start with two division algorithms for polynomials
of a single variable x; long division and synthetic
division.
The questions of factorability and finding roots of
real polynomials (polynomials with real coefficients)
are the main themes of this chapter.
Long division of polynomials is analogous to the
long division of numbers and it’s a general division
method for P(x) ÷ D(x).
Synthetic division is a simpler method for dividing
polynomials P(x) by monomials of the form (x – c),
i.e. P(x) ÷ (x – c). Synthetic division is particularly
useful for checking possible roots or finding
remainders of the division.

7. The Long Division
D(x)
P(x)
Dividend
Divisor

8. The Long Division
Set up for the division P(x) ÷ D(x) the
same way as for dividing numbers.D(x)
P(x)
Dividend
Divisor

9. The Long Division
Example A. Divide using long division.
Set up for the division P(x) ÷ D(x) the
same way as for dividing numbers.
x – 4
2x2 – 3x + 4
D(x)
P(x)
Dividend
Divisor

10. The Long Division
Example A. Divide using long division.
Set up for the division P(x) ÷ D(x) the
same way as for dividing numbers.
x – 4
2x2 – 3x + 4
1. Set up the long division
D(x)
P(x)
Dividend
Divisor

11. The Long Division
Example A. Divide using long division.
Set up for the division P(x) ÷ D(x) the
same way as for dividing numbers.
x – 4
2x2 – 3x + 4
1. Set up the long division
2x2 – 3x + 4x – 4
D(x)
P(x)
Dividend
Divisor

12. The Long Division
Example A. Divide using long division.
Set up for the division P(x) ÷ D(x) the
same way as for dividing numbers.
x – 4
2x2 – 3x + 4
1. Set up the long division
2x2 – 3x + 4x – 4
D(x)
P(x)
Dividend
Divisor
2. Enter on top the quotient
of the leading terms .

13. The Long Division
Example A. Divide using long division.
Set up for the division P(x) ÷ D(x) the
same way as for dividing numbers.
x – 4
2x2 – 3x + 4
1. Set up the long division
2x2 – 3x + 4x – 4
2. Enter on top the quotient
of the leading terms .
2x
D(x)
P(x)
Dividend
Divisor

14. The Long Division
Example A. Divide using long division.
Set up for the division P(x) ÷ D(x) the
same way as for dividing numbers.
x – 4
2x2 – 3x + 4
1. Set up the long division
2x2 – 3x + 4x – 4
2x
3. Multiply this quotient to the
divisor and subtract the result
from the dividend.
D(x)
P(x)
Dividend
Divisor
2. Enter on top the quotient
of the leading terms .

15. The Long Division
Example A. Divide using long division.
Set up for the division P(x) ÷ D(x) the
same way as for dividing numbers.
x – 4
2x2 – 3x + 4
1. Set up the long division
2x2 – 3x + 4x – 4
2x
2x2 – 8x3. Multiply this quotient to the
divisor and subtract the result
from the dividend.
– )
D(x)
P(x)
Dividend
Divisor
2. Enter on top the quotient
of the leading terms .

16. The Long Division
Example A. Divide using long division.
x – 4
2x2 – 3x + 4
1. Set up the long division
2x2 – 3x + 4x – 4
2x
2x2 – 8x3. Multiply this quotient to the
divisor and subtract the result
from the dividend.
– )
– +
Set up for the division P(x) ÷ D(x) the
same way as for dividing numbers.D(x)
P(x)
Dividend
Divisor
2. Enter on top the quotient
of the leading terms .
change the
signs then add

17. The Long Division
Example A. Divide using long division.
x – 4
2x2 – 3x + 4
1. Set up the long division
2x2 – 3x + 4x – 4
2x
2x2 – 8x3. Multiply this quotient to the
divisor and subtract the result
from the dividend.
– )
– +
5x + 4
Set up for the division P(x) ÷ D(x) the
same way as for dividing numbers.D(x)
P(x)
Dividend
Divisor
2. Enter on top the quotient
of the leading terms .

18. The Long Division
Example A. Divide using long division.
x – 4
2x2 – 3x + 4
1. Set up the long division
2x2 – 3x + 4x – 4
2x
2x2 – 8x3. Multiply this quotient to the
divisor and subtract the result
from the dividend.
– )
– +
5x + 4
Set up for the division P(x) ÷ D(x) the
same way as for dividing numbers.D(x)
P(x)
Dividend
Divisor
new
dividend
2. Enter on top the quotient
of the leading terms .
4. Use this difference from the
subtraction as the new dividend,
repeat steps 2 and 3.

19. The Long Division
Example A. Divide using long division.
x – 4
2x2 – 3x + 4
1. Set up the long division
2x2 – 3x + 4x – 4
2x + 5
2x2 – 8x3. Multiply this quotient to the
divisor and subtract the result
from the dividend.
4. Use this difference from the
subtraction as the new dividend,
repeat steps 2 and 3.
– )
– +
5x + 4
Set up for the division P(x) ÷ D(x) the
same way as for dividing numbers.D(x)
P(x)
Dividend
Divisor
new
dividend
2. Enter on top the quotient
of the leading terms .

20. The Long Division
Example A. Divide using long division.
x – 4
2x2 – 3x + 4
1. Set up the long division
2x2 – 3x + 4x – 4
2x + 5
2x2 – 8x3. Multiply this quotient to the
divisor and subtract the result
from the dividend.
4. Use this difference from the
subtraction as the new dividend,
repeat steps 2 and 3.
– )
– +
5x + 4
5x – 20– )
Set up for the division P(x) ÷ D(x) the
same way as for dividing numbers.D(x)
P(x)
Dividend
Divisor
new
dividend
2. Enter on top the quotient
of the leading terms .

21. The Long Division
Example A. Divide using long division.
x – 4
2x2 – 3x + 4
1. Set up the long division
2x2 – 3x + 4x – 4
2x + 5
2x2 – 8x3. Multiply this quotient to the
divisor and subtract the result
from the dividend.
4. Use this difference from the
subtraction as the new dividend,
repeat steps 2 and 3.
– )
– +
5x + 4
5x – 20– )
– +
24
Set up for the division P(x) ÷ D(x) the
same way as for dividing numbers.D(x)
P(x)
Dividend
Divisor
new
dividend
2. Enter on top the quotient
of the leading terms .

22. The Long Division
Example A. Divide using long division.
x – 4
2x2 – 3x + 4
1. Set up the long division
2x2 – 3x + 4x – 4
2x + 5
2x2 – 8x3. Multiply this quotient to the
divisor and subtract the result
from the dividend.
4. Use this difference from the
subtraction as the new dividend,
repeat steps 2 and 3.
5. Stop when the degree of the new dividend is smaller than
the degree of the divisor, i.e. no more quotient is possible.
– )
– +
5x + 4
5x – 20– )
– +
24
Set up for the division P(x) ÷ D(x) the
same way as for dividing numbers.D(x)
P(x)
Dividend
Divisor
new
dividend
2. Enter on top the quotient
of the leading terms .

23. The Long Division
2x2 – 3x + 4x – 4
2x + 5
2x2 – 8x
5x + 4
5x – 20
24
Here are the names of the terms:
dividend P(x)divisor D(x)
quotient Q(x)
remainder R(x)

24. The Long Division
2x2 – 3x + 4x – 4
2x + 5
2x2 – 8x
5x + 4
5x – 20
24
Here are the names of the terms:
dividend P(x)divisor D(x)
quotient Q(x)
remainder R(x)
We check easily that
x – 4
2x2 – 3x + 4
= 2x + 5 + x – 4
24

25. The Long Division
2x2 – 3x + 4x – 4
2x + 5
2x2 – 8x
5x + 4
5x – 20
24
Here are the names of the terms:
dividend P(x)divisor D(x)
quotient Q(x)
remainder R(x)
We check easily that
x – 4
2x2 – 3x + 4
= 2x + 5 + x – 4
24
i.e. = Q +
P
D
R
D

26. The Long Division
2x2 – 3x + 4x – 4
2x + 5
2x2 – 8x
5x + 4
5x – 20
24
Here are the names of the terms:
dividend P(x)divisor D(x)
quotient Q(x)
remainder R(x)
We check easily that
x – 4
2x2 – 3x + 4
= 2x + 5 + x – 4
24
We summarize the end result from performing the
long division algorithm in the following theorem.
i.e. = Q +
P
D
R
D

27. The Long Division
Long Division Theorem: Using long division for
P(x) ÷ D(x), we may obtain a quotient Q(x) and a
reminder R(x) such that

28. The Long Division
Long Division Theorem: Using long division for
P(x) ÷ D(x), we may obtain a quotient Q(x) and a
reminder R(x) such that
D(x)
P(x) = Q(x) +
D(x)
R(x) with deg R(x) < deg D(x).

29. The Long Division
Long Division Theorem: Using long division for
P(x) ÷ D(x), we may obtain a quotient Q(x) and a
reminder R(x) such that
D(x)
P(x) = Q(x) +
D(x)
R(x) with deg R(x) < deg D(x).
Example B. Divide
x2 + 1
x3 – 2x2 + 3
and write it in the
form of Q(x) +
D(x)
R(x)

30. The Long Division
Long Division Theorem: Using long division for
P(x) ÷ D(x), we may obtain a quotient Q(x) and a
reminder R(x) such that
D(x)
P(x) = Q(x) +
D(x)
R(x) with deg R(x) < deg D(x).
Example B. Divide
x2 + 1
x3 – 2x2 + 3
and write it in the
form of
x3 – 2x2 + 0x + 3x2 + 1
Q(x) +
D(x)
R(x)

31. The Long Division
Long Division Theorem: Using long division for
P(x) ÷ D(x), we may obtain a quotient Q(x) and a
reminder R(x) such that
D(x)
P(x) = Q(x) +
D(x)
R(x) with deg R(x) < deg D(x).
Example B. Divide
x2 + 1
x3 – 2x2 + 3
and write it in the
form of
x3 – 2x2 + 0x + 3x2 + 1
x
Q(x) +
D(x)
R(x)

32. The Long Division
Long Division Theorem: Using long division for
P(x) ÷ D(x), we may obtain a quotient Q(x) and a
reminder R(x) such that
D(x)
P(x) = Q(x) +
D(x)
R(x) with deg R(x) < deg D(x).
Example B. Divide
x2 + 1
x3 – 2x2 + 3
and write it in the
form of
x3 – 2x2 + 0x + 3x2 + 1
x
x3 + x– )
Q(x) +
D(x)
R(x)

33. The Long Division
Long Division Theorem: Using long division for
P(x) ÷ D(x), we may obtain a quotient Q(x) and a
reminder R(x) such that
D(x)
P(x) = Q(x) +
D(x)
R(x) with deg R(x) < deg D(x).
Example B. Divide
x2 + 1
x3 – 2x2 + 3
and write it in the
form of
x3 – 2x2 + 0x + 3x2 + 1
x
x3 + x– )
––
– 2x2 – x + 3
Q(x) +
D(x)
R(x)

34. The Long Division
Long Division Theorem: Using long division for
P(x) ÷ D(x), we may obtain a quotient Q(x) and a
reminder R(x) such that
D(x)
P(x) = Q(x) +
D(x)
R(x) with deg R(x) < deg D(x).
Example B. Divide
x2 + 1
x3 – 2x2 + 3
and write it in the
form of
x3 – 2x2 + 0x + 3x2 + 1
x – 2
x3 + x– )
––
– 2x2 – x + 3
Q(x) +
D(x)
R(x)

35. The Long Division
Long Division Theorem: Using long division for
P(x) ÷ D(x), we may obtain a quotient Q(x) and a
reminder R(x) such that
D(x)
P(x) = Q(x) +
D(x)
R(x) with deg R(x) < deg D(x).
Example B. Divide
x2 + 1
x3 – 2x2 + 3
and write it in the
form of
x3 – 2x2 + 0x + 3x2 + 1
x – 2
x3 + x– )
–
– )
–
– 2x2 – x + 3
Q(x) +
D(x)
R(x) – 2x2 – 2

36. The Long Division
Long Division Theorem: Using long division for
P(x) ÷ D(x), we may obtain a quotient Q(x) and a
reminder R(x) such that
D(x)
P(x) = Q(x) +
D(x)
R(x) with deg R(x) < deg D(x).
Example B. Divide
x2 + 1
x3 – 2x2 + 3
and write it in the
form of
x3 – 2x2 + 0x + 3x2 + 1
x – 2
x3 + x– )
–
– )
+
– x + 5
–
– 2x2 – x + 3
– 2x2 – 2Q(x) +
D(x)
R(x) +

37. The Long Division
Long Division Theorem: Using long division for
P(x) ÷ D(x), we may obtain a quotient Q(x) and a
reminder R(x) such that
D(x)
P(x) = Q(x) +
D(x)
R(x) with deg R(x) < deg D(x).
Example B. Divide
x2 + 1
x3 – 2x2 + 3
and write it in the
form of
x3 – 2x2 + 0x + 3x2 + 1
x – 2
x3 + x– )
–
– )
+
– x + 5
–
– 2x2 – x + 3
Q(x) +
D(x)
R(x) +
Stop!
The degree of R is less than
the degree of D
– 2x2 – 2

38. The Long Division
Long Division Theorem: Using long division for
P(x) ÷ D(x), we may obtain a quotient Q(x) and a
reminder R(x) such that
D(x)
P(x) = Q(x) +
D(x)
R(x) with deg R(x) < deg D(x).
Example B. Divide
x2 + 1
x3 – 2x2 + 3
and write it in the
form of
x3 – 2x2 + 0x + 3x2 + 1
x – 2
x3 + x– )
–
– )
++
– x + 5
–
– 2x2 – x + 3
Q(x) +
D(x)
R(x)
Hence
x2 + 1
x3 – 2x2 + 3
= x – 2 +
x2 + 1
– x + 5
– 2x2 – 2

39. Synthetic Division
When the divisor is of the form x – c, we may use a
simpler procedure called synthetic division.

40. Synthetic Division
When the divisor is of the form x – c, we may use a
simpler procedure called synthetic division.
Example C. Divide using synthetic
division.
x – 4
2x2 – 3x + 4

41. Synthetic Division
When the divisor is of the form x – c, we may use a
simpler procedure called synthetic division.
Example C. Divide using synthetic
division.
x – 4
2x2 – 3x + 4
1. To set up the synthetic division, list the
coefficients of the dividend in descending
order,

42. Synthetic Division
When the divisor is of the form x – c, we may use a
simpler procedure called synthetic division.
Example C. Divide using synthetic
division.
x – 4
2x2 – 3x + 4
2 –3 4
1. To set up the synthetic division, list the
coefficients of the dividend in descending
degree order,

43. Synthetic Division
When the divisor is of the form x – c, we may use a
simpler procedure called synthetic division.
Example C. Divide using synthetic
division.
x – 4
2x2 – 3x + 4
2 –3 4
1. To set up the synthetic division, list the
coefficients of the dividend in descending
degree order, put the number c to the left,
draw a line below them as shown.

44. Synthetic Division
When the divisor is of the form x – c, we may use a
simpler procedure called synthetic division.
Example C. Divide using synthetic
division.
x – 4
2x2 – 3x + 4
2 –3 44
1. To set up the synthetic division, list the
coefficients of the dividend in descending
degree order, put the number c to the left,
draw a line below them as shown.

45. Synthetic Division
When the divisor is of the form x – c, we may use a
simpler procedure called synthetic division.
Example C. Divide using synthetic
division.
x – 4
2x2 – 3x + 4
2. To "divide", bring down the leading
coefficient,
2 –3 44
2
1. To set up the synthetic division, list the
coefficients of the dividend in descending
degree order, put the number c to the left,
draw a line below them as shown.

46. Synthetic Division
When the divisor is of the form x – c, we may use a
simpler procedure called synthetic division.
Example C. Divide using synthetic
division.
x – 4
2x2 – 3x + 4
1. To set up the synthetic division, list the
coefficients of the dividend in descending
degree order, put the number c to the left,
draw a line below them as shown.
2. To "divide", bring down the leading
coefficient, multiply it by c and enter the
result in next column.
2 –3 44
2

47. Synthetic Division
When the divisor is of the form x – c, we may use a
simpler procedure called synthetic division.
Example C. Divide using synthetic
division.
x – 4
2x2 – 3x + 4
1. To set up the synthetic division, list the
coefficients of the dividend in descending
degree order, put the number c to the left,
draw a line below them as shown.
2. To "divide", bring down the leading
coefficient, multiply it by c and enter the
result in next column.
2 –3 44
2
8

48. Synthetic Division
When the divisor is of the form x – c, we may use a
simpler procedure called synthetic division.
Example C. Divide using synthetic
division.
x – 4
2x2 – 3x + 4
1. To set up the synthetic division, list the
coefficients of the dividend in descending
degree order, put the number c to the left,
draw a line below them as shown.
2. To "divide", bring down the leading
coefficient, multiply it by c and enter the
result in next column. Add the column to
get the sum.
2 –3 44
2
8
5

49. Synthetic Division
When the divisor is of the form x – c, we may use a
simpler procedure called synthetic division.
Example C. Divide using synthetic
division.
x – 4
2x2 – 3x + 4
1. To set up the synthetic division, list the
coefficients of the dividend in descending
degree order, put the number c to the left,
draw a line below them as shown.
2. To "divide", bring down the leading
coefficient, multiply it by c and enter the
result in next column. Add the column to
get the sum. Multiply the sum by c, enter
the result in the next column then add.
2 –3 44
2
8
5

50. Synthetic Division
When the divisor is of the form x – c, we may use a
simpler procedure called synthetic division.
Example C. Divide using synthetic
division.
x – 4
2x2 – 3x + 4
1. To set up the synthetic division, list the
coefficients of the dividend in descending
degree order, put the number c to the left,
draw a line below them as shown.
2. To "divide", bring down the leading
coefficient, multiply it by c and enter the
result in next column. Add the column to
get the sum. Multiply the sum by c, enter
the result in the next column then add.
2 –3 44
2
8
5
20

51. Synthetic Division
When the divisor is of the form x – c, we may use a
simpler procedure called synthetic division.
Example C. Divide using synthetic
division.
x – 4
2x2 – 3x + 4
1. To set up the synthetic division, list the
coefficients of the dividend in descending
degree order, put the number c to the left,
draw a line below them as shown.
2 –3 44
2
8
5
202. To "divide", bring down the leading
coefficient, multiply it by c and enter the
result in next column. Add the column to
get the sum. Multiply the sum by c, enter
the result in the next column then add.
Continue this process to the last column
and the procedure stops.

52. Synthetic Division
When the divisor is of the form x – c, we may use a
simpler procedure called synthetic division.
Example C. Divide using synthetic
division.
x – 4
2x2 – 3x + 4
1. To set up the synthetic division, list the
coefficients of the dividend in descending
degree order, put the number c to the left,
draw a line below them as shown.
2 –3 44
2
8
5
202. To "divide", bring down the leading
coefficient, multiply it by c and enter the
result in next column. Add the column to
get the sum. Multiply the sum by c, enter
the result in the next column then add.
Continue this process to the last column
and the procedure stops.
24
Add and the
procedure stops.

53. Synthetic Division
2 –3 44
2
8
5
20
The result of the division is read in the bottom row.
24
The last number
is the remainder,
it’s 24.

54. Synthetic Division
2 –3 44
2
8
5
20
The result of the division is read in the bottom row.
24
The last number
is the remainder,
it’s 24.
these numbers are the
coefficients of the quotient
polynomial which is one
degree less than the
dividend, it’s 2x + 5.

55. Synthetic Division
2 –3 44
2
8
5
20
The result of the division is read in the bottom row.
24
The last number
is the remainder,
it’s 24.
these numbers are the
coefficients of the quotient
polynomial which is one
degree less than the
dividend, it's 2x + 5.
Hence
x – 4
2x2 – 3x + 4 = 2x + 5 +
x – 4
24

56. Synthetic Division
Example C. Divide
using synthetic division.
x + 2
2x5 – 7x3 – 3x + 2

57. Synthetic Division
Example C. Divide
using synthetic division.
x + 2
2x5 – 7x3 – 3x + 2
Set up the division, make sure to put 0 for the
missing x-terms.

58. Synthetic Division
Example C. Divide
using synthetic division.
x + 2
2x5 – 7x3 – 3x + 2
Set up the division, make sure to put 0 for the
missing x-terms.
2 0 –7 0 –3 2

59. Synthetic Division
Example C. Divide
using synthetic division.
x + 2
2x5 – 7x3 – 3x + 2
Set up the division, make sure to put 0 for the
missing x-terms. Since x + 2 = x – (–2), we use
x = –2 for the division.
2 0 –7 0 –3 2–2

60. Synthetic Division
Example C. Divide
using synthetic division.
x + 2
2x5 – 7x3 – 3x + 2
Set up the division, make sure to put 0 for the
missing x-terms. Since x + 2 = x – (–2), we use
x = –2 for the division.
2 0 –7 0 –3 2–2
2

61. Synthetic Division
Example C. Divide
using synthetic division.
x + 2
2x5 – 7x3 – 3x + 2
Set up the division, make sure to put 0 for the
missing x-terms. Since x + 2 = x – (–2), we use
x = –2 for the division.
2 0 –7 0 –3 2–2
2
–4multiply

62. Synthetic Division
Example C. Divide
using synthetic division.
x + 2
2x5 – 7x3 – 3x + 2
Set up the division, make sure to put 0 for the
missing x-terms. Since x + 2 = x – (–2), we use
x = –2 for the division.
2 0 –7 0 –3 2–2
2
–4
add
–4

63. Synthetic Division
Example C. Divide
using synthetic division.
x + 2
2x5 – 7x3 – 3x + 2
Set up the division, make sure to put 0 for the
missing x-terms. Since x + 2 = x – (–2), we use
x = –2 for the division.
2 0 –7 0 –3 2–2
2
–4multiply
–4
8

64. Synthetic Division
Example C. Divide
using synthetic division.
x + 2
2x5 – 7x3 – 3x + 2
Set up the division, make sure to put 0 for the
missing x-terms. Since x + 2 = x – (–2), we use
x = –2 for the division.
2 0 –7 0 –3 2–2
2
–4
–4
8
1

65. Synthetic Division
Example C. Divide
using synthetic division.
x + 2
2x5 – 7x3 – 3x + 2
Set up the division, make sure to put 0 for the
missing x-terms. Since x + 2 = x – (–2), we use
x = –2 for the division.
2 0 –7 0 –3 2–2
2
–4
–4
8
1
–2
–2

66. Synthetic Division
Example C. Divide
using synthetic division.
x + 2
2x5 – 7x3 – 3x + 2
Set up the division, make sure to put 0 for the
missing x-terms. Since x + 2 = x – (–2), we use
x = –2 for the division.
2 0 –7 0 –3 2–2
2
–4
–4
8
1
–2
–2
4
1

67. Synthetic Division
Example C. Divide
using synthetic division.
x + 2
2x5 – 7x3 – 3x + 2
Set up the division, make sure to put 0 for the
missing x-terms. Since x + 2 = x – (–2), we use
x = –2 for the division.
2 0 –7 0 –3 2–2
2
–4
–4
8
1
–2
–2
4
1
–2
0

68. Synthetic Division
Example C. Divide
using synthetic division.
x + 2
2x5 – 7x3 – 3x + 2
Set up the division, make sure to put 0 for the
missing x-terms. Since x + 2 = x – (–2), we use
x = –2 for the division.
2 0 –7 0 –3 2–2
2
–4
–4
8
1
–2
–2
4
1
–2
0
So
x + 2
2x5 – 7x3 – 3x + 2 = 2x4 – 4x3 + x2 – 2x + 1

69. Synthetic Division
Example C. Divide
using synthetic division.
x + 2
2x5 – 7x3 – 3x + 2
Set up the division, make sure to put 0 for the
missing x-terms. Since x + 2 = x – (–2), we use
x = –2 for the division.
2 0 –7 0 –3 2–2
2
–4
–4
8
1
–2
–2
4
1
–2
0
So
x + 2
2x5 – 7x3 – 3x + 2 = 2x4 – 4x3 + x2 – 2x + 1
Note that because the remainder is 0, we have that
2x5 – 7x3 – 3x + 2 = (x + 2) (2x4 – 4x3 + x2 – 2x + 1)
and that x = –2 is a root.

70. Long Division
Exercise A. Divide P(x) ÷ D(x) using long division,
D(x)
P(x)
as Q(x)+
D(x)
R(x)
with deg R(x) < deg D(x).
1. x + 3
–2x + 3
and write
4. x + 3
x2 – 9
7.
x + 3
x2 – 2x + 3
2. x + 1
3x + 2 3. 2x – 1
3x + 1
8.
x – 3
2x2 – 2x + 1 9.
2x + 1
–2x2 + 4x + 1
5. x + 2
x2 + 4
6. x – 3
x2 + 9
10.
x + 3
x3 – 2x + 3 11.
x – 3
2x3 – 2x + 1 12.
2x + 1
–2x3 + 4x + 1
13.
x2 + x + 3
x3 – 2x + 3 14.
x2 – 3
2x3 – 2x + 1 15.
x2 – 2x + 1
–2x3 + 4x + 1
16.
x – 1
x30 – 2x20+ 1
16.
x + 1
x30 – 2x20 + 1 18.
x – 1
xN – 1 (N > 1)
(Many of them can be done by synthetic division. )

71. Synthetic Division
B. Divide P(x) ÷ (x – c) using synthetic division,
D(x)
P(x)
as Q(x) + x – c
r where r is a number.
1. x + 3
–2x + 3
and write
2. x + 1
3x + 2 3. x – 2
3x + 1
4.
x + 3
x2 – 2x + 3 5.
x – 3
2x2 – 2x + 1 6.
x + 2
–2x2 + 4x + 1
7.
x + 3
x3 – 2x + 3 8.
x – 3
2x3 – 2x + 1 9.
x + 4
–2x3 + 4x + 1
10.
x – 1
x30 – 2x + 1
11.
x + 1
x30 – 2x20 + 1 12.
x – 1
xN – 1 (N > 1)
13. Use synthetic division to verify that
(x3 – 7x – 6) / (x + 2) divides completely with
remainder 0, then factor x3 – 7x – 6 completely.

72. The Long Division
(Answers to odd problems) Exercise A.
1. x + 3
9
7.
3. 2
3
9.
5.
11.
13. 15.
17. x29 – x28 + x27 – x26 + x25 – x24 + x23 – x22 + x21 – x20
– x19 + x18 – x17 + x16 – x15 + x14 – x13 + x12 – x11 + x10
– x9 + x8 – x7 + x6 – x5 + x4 – x3 + x2 – x2 + 1
–2 + + 2(2x – 1)
5
x + 2
8(x – 2) +
x +3
18(x – 5) +
(– x + )2
5 –
2(2x – 1)
5
x – 3
49(x2 + 6x +16) +
x2 + x + 3
2(2x – 3)
(x – 1) –
(x – 1)2
5 – 2x
–(2x + 4) +

73. The Long Division
Exercise B.
1. 3.
5. 7.
9.
11. x29 – x28 + x27 – x26 + x25 – x24 + x23 – x22 + x21
13. (x + 1)(x + 2)(x – 3)
x + 3
9–2 +
x – 2
73 +
x – 3
13(2x + 4) +
x + 3
18(x2 – 3x + 7) –
x + 4
113(– 2x2 + 8x – 28) +
– x20 – x19 + x18 – x17 + x16 – x15 + x14 – x13 + x12 – x11
+ x10 – x9 + x8 – x7 + x6 – x5 + x4 – x3 + x2 – x2 + 1