multiplication and division of rational expressions

1. Multiplication and Division of Rational Expressions
Frank Ma © 2011

2. Multiplication Rule for Rational Expressions
A
B
C
D
* =
AC
BD
Multiplication and Division of Rational Expressions

3. Multiplication Rule for Rational Expressions
A
B
C
D
* =
AC
BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the
top and the bottom, then cancel.

4. Multiplication Rule for Rational Expressions
A
B
C
D
* =
AC
BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the
top and the bottom, then cancel.
Example A. Simplify
10x
y3z
a. *
y2
5x3

5. Multiplication Rule for Rational Expressions
A
B
C
D
* =
AC
BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the
top and the bottom, then cancel.
Example A. Simplify
10x
y3z
a. *
y2
5x3 = 10xy2
5x3y3z

6. Multiplication Rule for Rational Expressions
A
B
C
D
* =
AC
BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the
top and the bottom, then cancel.
Example A. Simplify
10x
y3z
a. *
y2
5x3 = 10xy2
5x3y3z = 2
x2yz

7. Multiplication Rule for Rational Expressions
A
B
C
D
* =
AC
BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the
top and the bottom, then cancel.
Example A. Simplify
10x
y3z
a. *
y2
5x3 = 10xy2
5x3y3z = 2
x2yz
b.
(x2 + 2x – 3 )
(x – 2) (x2 – x )
(x2 – 4 )
*

8. Multiplication Rule for Rational Expressions
A
B
C
D
* =
AC
BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the
top and the bottom, then cancel.
Example A. Simplify
10x
y3z
a. *
y2
5x3 = 10xy2
5x3y3z = 2
x2yz
b.
(x2 + 2x – 3 )
(x – 2) (x2 – x )
(x2 – 4 )
*
=
(x + 3)(x – 1 )

9. Multiplication Rule for Rational Expressions
A
B
C
D
* =
AC
BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the
top and the bottom, then cancel.
Example A. Simplify
10x
y3z
a. *
y2
5x3 = 10xy2
5x3y3z = 2
x2yz
b.
(x2 + 2x – 3 )
(x – 2) (x2 – x )
(x2 – 4 )
*
=
(x + 3)(x – 1 ) (x + 2 )(x – 2)

10. Multiplication Rule for Rational Expressions
A
B
C
D
* =
AC
BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the
top and the bottom, then cancel.
Example A. Simplify
10x
y3z
a. *
y2
5x3 = 10xy2
5x3y3z = 2
x2yz
b.
(x2 + 2x – 3 )
(x – 2) (x2 – x )
(x2 – 4 )
*
=
(x + 3)(x – 1 )
(x – 2)
(x + 2 )(x – 2)

11. Multiplication Rule for Rational Expressions
A
B
C
D
* =
AC
BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the
top and the bottom, then cancel.
Example A. Simplify
10x
y3z
a. *
y2
5x3 = 10xy2
5x3y3z = 2
x2yz
b.
(x2 + 2x – 3 )
(x – 2) (x2 – x )
(x2 – 4 )
*
=
(x + 3)(x – 1 )
(x – 2) x(x – 1 )
(x + 2 )(x – 2)

12. Multiplication Rule for Rational Expressions
A
B
C
D
* =
AC
BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the
top and the bottom, then cancel.
Example A. Simplify
10x
y3z
a. *
y2
5x3 = 10xy2
5x3y3z = 2
x2yz
b.
(x2 + 2x – 3 )
(x – 2) (x2 – x )
(x2 – 4 )
*
=
(x + 3)(x – 1 )
(x – 2) x(x – 1 )
(x + 2 )(x – 2)

13. Multiplication Rule for Rational Expressions
A
B
C
D
* =
AC
BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the
top and the bottom, then cancel.
Example A. Simplify
10x
y3z
a. *
y2
5x3 = 10xy2
5x3y3z = 2
x2yz
b.
(x2 + 2x – 3 )
(x – 2) (x2 – x )
(x2 – 4 )
*
=
(x + 3)(x – 1 )
(x – 2) x(x – 1 )
(x + 2 )(x – 2)

14. Multiplication Rule for Rational Expressions
A
B
C
D
* =
AC
BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the
top and the bottom, then cancel.
Example A. Simplify
10x
y3z
a. *
y2
5x3 = 10xy2
5x3y3z = 2
x2yz
b.
(x2 + 2x – 3 )
(x – 2) (x2 – x )
(x2 – 4 )
=
(x + 3)(x + 2)
x
*
=
(x + 3)(x – 1 )
(x – 2) x(x – 1 )
(x + 2 )(x – 2)
In the next section, we meet the following type of problems.

15. Multiplication and Division of Rational Expressions
Example B. Simplify and expand the answers.
a. x + 3
x – 1
(x2 – 1)

16. Multiplication and Division of Rational Expressions
a. x + 3
x – 1
(x2 – 1)
= x + 3
(x – 1) (x – 1)(x + 1)
Example B. Simplify and expand the answers.

17. Multiplication and Division of Rational Expressions
Example B. Simplify and expand the answers.
a. x + 3
x – 1
(x2 – 1)
= x + 3
(x – 1) (x – 1)(x + 1)
= (x + 3)(x + 1)

18. Multiplication and Division of Rational Expressions
a. x + 3
x – 1
(x2 – 1)
= x + 3
(x – 1) (x – 1)(x + 1)
= (x + 3)(x + 1) = x2 + 4x + 3
Example B. Simplify and expand the answers.

19. Multiplication and Division of Rational Expressions
a. x + 3
x – 1
(x2 – 1)
= x + 3
(x – 1) (x – 1)(x + 1)
= (x + 3)(x + 1) = x2 + 4x + 3
b. x – 2
x2 – 9
( –
x + 1
x2 – 2x – 3
) ( x – 3)(x + 3)(x + 1)
Example B. Simplify and expand the answers.

20. Multiplication and Division of Rational Expressions
a. x + 3
x – 1
(x2 – 1)
= x + 3
(x – 1) (x – 1)(x + 1)
= (x + 3)(x + 1) = x2 + 4x + 3
b. x – 2
x2 – 9
( –
x + 1
x2 – 2x – 3
) ( x – 3)(x + 3)(x + 1)
=
x – 2
(x – 3)(x + 3)
Example B. Simplify and expand the answers.

21. Multiplication and Division of Rational Expressions
a. x + 3
x – 1
(x2 – 1)
= x + 3
(x – 1) (x – 1)(x + 1)
= (x + 3)(x + 1) = x2 + 4x + 3
b. x – 2
x2 – 9
( –
x + 1
x2 – 2x – 3
) ( x – 3)(x + 3)(x + 1)
=
x – 2
(x – 3)(x + 3)
–
x + 1
(x – 3)(x + 1)
Example B. Simplify and expand the answers.

22. Multiplication and Division of Rational Expressions
a. x + 3
x – 1
(x2 – 1)
= x + 3
(x – 1) (x – 1)(x + 1)
= (x + 3)(x + 1) = x2 + 4x + 3
b. x – 2
x2 – 9
( –
x + 1
x2 – 2x – 3
) ( x – 3)(x + 3)(x + 1)
=
x – 2
(x – 3)(x + 3)
[ –
x + 1
(x – 3)(x + 1)
] ( x – 3)(x + 3)(x + 1)
Example B. Simplify and expand the answers.

23. Multiplication and Division of Rational Expressions
a. x + 3
x – 1
(x2 – 1)
= x + 3
(x – 1) (x – 1)(x + 1)
= (x + 3)(x + 1) = x2 + 4x + 3
b. x – 2
x2 – 9
( –
x + 1
x2 – 2x – 3
) ( x – 3)(x + 3)(x + 1)
=
x – 2
(x – 3)(x + 3)
[ –
x + 1
(x – 3)(x + 1)
] ( x – 3)(x + 3)(x + 1)
(x + 1)
Example B. Simplify and expand the answers.

24. Multiplication and Division of Rational Expressions
a. x + 3
x – 1
(x2 – 1)
= x + 3
(x – 1) (x – 1)(x + 1)
= (x + 3)(x + 1) = x2 + 4x + 3
b. x – 2
x2 – 9
( –
x + 1
x2 – 2x – 3
) ( x – 3)(x + 3)(x + 1)
=
x – 2
(x – 3)(x + 3)
[ –
x + 1
(x – 3)(x + 1)
] ( x – 3)(x + 3)(x + 1)
(x + 1) (x + 3)
Example B. Simplify and expand the answers.

25. Multiplication and Division of Rational Expressions
a. x + 3
x – 1
(x2 – 1)
= x + 3
(x – 1) (x – 1)(x + 1)
= (x + 3)(x + 1) = x2 + 4x + 3
b. x – 2
x2 – 9
( –
x + 1
x2 – 2x – 3
) ( x – 3)(x + 3)(x + 1)
=
x – 2
(x – 3)(x + 3)
[ –
x + 1
(x – 3)(x + 1)
] ( x – 3)(x + 3)(x + 1)
(x + 1) (x + 3)
= (x – 2)(x + 1) – (x + 1)(x + 3)
Example B. Simplify and expand the answers.

26. Multiplication and Division of Rational Expressions
a. x + 3
x – 1
(x2 – 1)
= x + 3
(x – 1) (x – 1)(x + 1)
= (x + 3)(x + 1) = x2 + 4x + 3
b. x – 2
x2 – 9
( –
x + 1
x2 – 2x – 3
) ( x – 3)(x + 3)(x + 1)
=
x – 2
(x – 3)(x + 3)
[ –
x + 1
(x – 3)(x + 1)
] ( x – 3)(x + 3)(x + 1)
(x + 1) (x + 3)
= (x – 2)(x + 1) – (x + 1)(x + 3)
= (x – 2)(x + 1) + (–x –1)(x + 3)
Example B. Simplify and expand the answers.

27. Multiplication and Division of Rational Expressions
a. x + 3
x – 1
(x2 – 1)
= x + 3
(x – 1) (x – 1)(x + 1)
= (x + 3)(x + 1) = x2 + 4x + 3
b. x – 2
x2 – 9
( –
x + 1
x2 – 2x – 3
) ( x – 3)(x + 3)(x + 1)
=
x – 2
(x – 3)(x + 3)
[ –
x + 1
(x – 3)(x + 1)
] ( x – 3)(x + 3)(x + 1)
(x + 1) (x + 3)
= (x – 2)(x + 1) – (x + 1)(x + 3)
= (x – 2)(x + 1) + (–x –1)(x + 3)
= x2 – x – 2 – x2 – 4x – 3
Example B. Simplify and expand the answers.

28. Multiplication and Division of Rational Expressions
Example B. Simplify and expand the answers.
a. x + 3
x – 1
(x2 – 1)
= x + 3
(x – 1) (x – 1)(x + 1)
= (x + 3)(x + 1) = x2 + 4x + 3
b. x – 2
x2 – 9
( –
x + 1
x2 – 2x – 3
) ( x – 3)(x + 3)(x + 1)
=
x – 2
(x – 3)(x + 3)
[ –
x + 1
(x – 3)(x + 1)
] ( x – 3)(x + 3)(x + 1)
(x + 1) (x + 3)
= (x – 2)(x + 1) – (x + 1)(x + 3)
= (x – 2)(x + 1) + (–x –1)(x + 3)
= x2 – x – 2 – x2 – 4x – 3
= –5x – 5

29. Division Rule for Rational Expressions
Multiplication and Division of Rational Expressions
A
B
C
D
÷

30. Division Rule for Rational Expressions
Multiplication and Division of Rational Expressions
A
B
C
D
÷ =
AD
BC
Reciprocate

31. Division Rule for Rational Expressions
Multiplication and Division of Rational Expressions
A
B
C
D
÷ =
AD
BC
Reciprocate
We convert division by an expression of multiplying by its
reciprocal.

32. Division Rule for Rational Expressions
Multiplication and Division of Rational Expressions
A
B
C
D
÷ =
AD
BC
Reciprocate
We convert division by an expression of multiplying by its
reciprocal. Then we factor and reduce the product.

33. Division Rule for Rational Expressions
Multiplication and Division of Rational Expressions
A
B
C
D
÷ =
AD
BC
Reciprocate
(2x – 6)
(x + 3)
÷
(x2 + 2x – 3)
(9 – x2)
Example C. Simplify
We convert division by an expression of multiplying by its
reciprocal. Then we factor and reduce the product.

34. Division Rule for Rational Expressions
Multiplication and Division of Rational Expressions
A
B
C
D
÷ =
AD
BC
Reciprocate
(2x – 6)
(x + 3)
÷
(x2 + 2x – 3)
(9 – x2)
=
(2x – 6)
(x + 3)
(x2 + 2x – 3)
(9 – x2)
*
Example C. Simplify
We convert division by an expression of multiplying by its
reciprocal. Then we factor and reduce the product.

35. Division Rule for Rational Expressions
Multiplication and Division of Rational Expressions
A
B
C
D
÷ =
AD
BC
Reciprocate
(2x – 6)
(x + 3)
÷
(x2 + 2x – 3)
(9 – x2)
=
(2x – 6)
(x + 3)
(x2 + 2x – 3)
(9 – x2)
*
=
2(x – 3)
(x + 3)
Example C. Simplify
We convert division by an expression of multiplying by its
reciprocal. Then we factor and reduce the product.

36. Division Rule for Rational Expressions
Multiplication and Division of Rational Expressions
A
B
C
D
÷ =
AD
BC
Reciprocate
(2x – 6)
(x + 3)
÷
(x2 + 2x – 3)
(9 – x2)
=
(2x – 6)
(x + 3)
(x2 + 2x – 3)
(9 – x2)
*
=
2(x – 3)
(x + 3)
(x + 3)(x – 1)
(3 – x)(3 + x)
Example C. Simplify
We convert division by an expression of multiplying by its
reciprocal. Then we factor and reduce the product.

37. Division Rule for Rational Expressions
Multiplication and Division of Rational Expressions
A
B
C
D
÷ =
AD
BC
Reciprocate
Example C. Simplify
(2x – 6)
(x + 3)
÷
(x2 + 2x – 3)
(9 – x2)
=
(2x – 6)
(x + 3)
(x2 + 2x – 3)
(9 – x2)
*
=
2(x – 3)
(x + 3)
(x + 3)(x – 1)
(3 – x)(3 + x)
*
(9 – x2)
We convert division by an expression of multiplying by its
reciprocal. Then we factor and reduce the product.

38. Division Rule for Rational Expressions
Multiplication and Division of Rational Expressions
A
B
C
D
÷ =
AD
BC
Reciprocate
(2x – 6)
(x + 3)
÷
(x2 + 2x – 3)
(9 – x2)
=
(2x – 6)
(x + 3)
(x2 + 2x – 3)
(9 – x2)
*
=
2(x – 3)
(x + 3)
(x + 3)(x – 1)
(3 – x)(3 + x)
*
(–1)
Example C. Simplify
We convert division by an expression of multiplying by its
reciprocal. Then we factor and reduce the product.

39. Division Rule for Rational Expressions
Multiplication and Division of Rational Expressions
A
B
C
D
÷ =
AD
BC
Reciprocate
(2x – 6)
(x + 3)
÷
(x2 + 2x – 3)
(9 – x2)
=
(2x – 6)
(x + 3)
(x2 + 2x – 3)
(9 – x2)
*
=
2(x – 3)
(x + 3)
(x + 3)(x – 1)
(3 – x)(3 + x)
*
(–1)
=
–2(x – 1)
(3 + x)
Example C. Simplify
We convert division by an expression of multiplying by its
reciprocal. Then we factor and reduce the product.

40. Multiplication and Division of Rational Expressions
Besides the expanded form and factored forms, rational
expressions may also be split into sums or differences.

41. Multiplication and Division of Rational Expressions
Besides the expanded form and factored forms, rational
expressions may also be split into sums or differences.
There are two common ways to do this.

42. Multiplication and Division of Rational Expressions
Besides the expanded form and factored forms, rational
expressions may also be split into sums or differences.
There are two common ways to do this.
I. Split off the numerator term by term.

43. Multiplication and Division of Rational Expressions
Example D. Break up the numerators as the sums or
differences and simplify each term.
(2x – 6)
(x + 3)
a. =
Besides the expanded form and factored forms, rational
expressions may also be split into sums or differences.
There are two common ways to do this.
I. Split off the numerator term by term.
(2x – 6)
3x2
b. =

44. Multiplication and Division of Rational Expressions
Example D. Break up the numerators as the sums or
differences and simplify each term.
(2x – 6)
(x + 3)
a. =
Besides the expanded form and factored forms, rational
expressions may also be split into sums or differences.
There are two common ways to do this.
I. Split off the numerator term by term.
(x + 3) – (x + 3)
2x 6
(2x – 6)
3x2
b. =

45. Multiplication and Division of Rational Expressions
Example D. Break up the numerators as the sums or
differences and simplify each term.
(2x – 6)
(x + 3)
a. =
Besides the expanded form and factored forms, rational
expressions may also be split into sums or differences.
There are two common ways to do this.
I. Split off the numerator term by term.
(x + 3) – (x + 3)
2x 6
(2x – 6)
3x2
b. = –
2x 6
3x2 3x2

46. Multiplication and Division of Rational Expressions
Example D. Break up the numerators as the sums or
differences and simplify each term.
(2x – 6)
(x + 3)
a. =
Besides the expanded form and factored forms, rational
expressions may also be split into sums or differences.
There are two common ways to do this.
I. Split off the numerator term by term.
(x + 3) – (x + 3)
2x 6
(2x – 6)
3x2
b. = –
2x 6
3x2 3x2 = –
2
x2
2
3x

47. Multiplication and Division of Rational Expressions
Example D. Break up the numerators as the sums or
differences and simplify each term.
(2x – 6)
(x + 3)
a. =
Besides the expanded form and factored forms, rational
expressions may also be split into sums or differences.
There are two common ways to do this.
I. Split off the numerator term by term.
(x + 3) – (x + 3)
2x 6
(2x – 6)
3x2
b. = –
2x 6
3x2 3x2 = –
2
x2
2
3x
II. Long Division
Long division is the extension of the long division of numbers
from grade school and it is for the division of polynomials in
one variable.

48. Multiplication and Division of Rational Expressions
Example D. Break up the numerators as the sums or
differences and simplify each term.
(2x – 6)
(x + 3)
a. =
Besides the expanded form and factored forms, rational
expressions may also be split into sums or differences.
There are two common ways to do this.
I. Split off the numerator term by term.
(x + 3) – (x + 3)
2x 6
(2x – 6)
3x2
b. = –
2x 6
3x2 3x2 = –
2
x2
2
3x
II. Long Division
Long division is the extension of the long division of numbers
from grade school and it is for the division of polynomials in
one variable. Specifically, long division gives relevant results
only when the degree of the numerator is the same or more
than the degree of the denominator.

49. Multiplication and Division of Rational Expressions
Let’s look at the example 125/8 or 125 ÷ 8 by long division.

50. Multiplication and Division of Rational Expressions
Let’s look at the example 125/8 or 125 ÷ 8 by long division.
i. Put the problem in the long division
format with the “bottom-out” and
move from left to right until there is
enough to enter a quotient.

51. Multiplication and Division of Rational Expressions
Let’s look at the example 125/8 or 125 ÷ 8 by long division.
i. Put the problem in the long division
format with the “bottom-out” and
move from left to right until there is
enough to enter a quotient. )8 125

52. Multiplication and Division of Rational Expressions
Let’s look at the example 125/8 or 125 ÷ 8 by long division.
i. Put the problem in the long division
format with the “bottom-out” and
move from left to right until there is
enough to enter a quotient. )8 125
1

53. Multiplication and Division of Rational Expressions
Let’s look at the example 125/8 or 125 ÷ 8 by long division.
i. Put the problem in the long division
format with the “bottom-out” and
move from left to right until there is
enough to enter a quotient. )8 125
1
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
8
45

54. Multiplication and Division of Rational Expressions
Let’s look at the example 125/8 or 125 ÷ 8 by long division.
i. Put the problem in the long division
format with the “bottom-out” and
move from left to right until there is
enough to enter a quotient. )8 125
1
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
8
45
iii. Repeat steps i and ii until no more
quotient may be entered.

55. Multiplication and Division of Rational Expressions
i. Put the problem in the long division
format with the “bottom-out” and
move from left to right until there is
enough to enter a quotient. )8 125
15
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
8
45
iii. Repeat steps i and ii until no more
quotient may be entered.
40
5
Let’s look at the example 125/8 or 125 ÷ 8 by long division.

56. Multiplication and Division of Rational Expressions
i. Put the problem in the long division
format with the “bottom-out” and
move from left to right until there is
enough to enter a quotient. )8 125
15
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
8
45
iii. Repeat steps i and ii until no more
quotient may be entered. Then we may
put the fraction N/D in the following
mixed form:
N
D
= Q + R
D
and that R (the remainder) is smaller then D (no more quotient).
40
5
where Q is the quotient
Let’s look at the example 125/8 or 125 ÷ 8 by long division.

57. Multiplication and Division of Rational Expressions
i. Put the problem in the long division
format with the “bottom-out” and
move from left to right until there is
enough to enter a quotient. )8 125
15
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
8
45
iii. Repeat steps i and ii until no more
quotient may be entered. Then we may
put the fraction N/D in the following
mixed form:
N
D
= Q + R
D
and that R (the remainder) is smaller then D (no more quotient).
40
5
125
8
= 15 + 5
8
where Q is the quotient
Let’s look at the example 125/8 or 125 ÷ 8 by long division.

58. Multiplication and Division of Rational Expressions
Example E. Divide using long division(2x – 6)
(x + 3)

59. Multiplication and Division of Rational Expressions
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
Example E. Divide using long division(2x – 6)
(x + 3)

60. Multiplication and Division of Rational Expressions
)x + 3 2x – 6
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
Make sure the terms
are in order.
Example E. Divide using long division(2x – 6)
(x + 3)

61. Multiplication and Division of Rational Expressions
)x + 3 2x – 6
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
Make sure the terms
are in order.
Example E. Divide using long division(2x – 6)
(x + 3)

62. Multiplication and Division of Rational Expressions
)x + 3 2x – 6
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
enter the quotients of the
leading terms 2x/x = 2
Example E. Divide using long division(2x – 6)
(x + 3)

63. Multiplication and Division of Rational Expressions
)x + 3 2x – 6
2
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
enter the quotients of the
leading terms 2x/x = 2
Example E. Divide using long division(2x – 6)
(x + 3)

64. Multiplication and Division of Rational Expressions
)x + 3 2x – 6
2
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
Example E. Divide using long division(2x – 6)
(x + 3)

65. Multiplication and Division of Rational Expressions
)x + 3 2x – 6
2
2x + 6
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
Example E. Divide using long division(2x – 6)
(x + 3)

66. Multiplication and Division of Rational Expressions
)x + 3 2x – 6
2
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
2x + 6
–12
–)
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
Example E. Divide using long division(2x – 6)
(x + 3)

67. Multiplication and Division of Rational Expressions
)x + 3 2x – 6
2
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
2x + 6
–12
iii. Repeat steps i and ii until no more
quotient may be entered.
–)
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
Example E. Divide using long division(2x – 6)
(x + 3)

68. Multiplication and Division of Rational Expressions
)x + 3 2x – 6
2
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
2x + 6
–12
–)
Stop. No more
quotient since
x can’t going into 12.
iii. Repeat steps i and ii until no more
quotient may be entered.
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
Example E. Divide using long division(2x – 6)
(x + 3)

69. Multiplication and Division of Rational Expressions
)x + 3 2x – 6
2
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
2x + 6
–12
iii. Repeat steps i and ii until no more
quotient may be entered. Then we may
put the fraction N/D in the following
mixed form:
–)
N
D
= Q + R
D
has smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
Example E. Divide using long division(2x – 6)
(x + 3)

70. Multiplication and Division of Rational Expressions
)x + 3 2x – 6
2
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
2x + 6
–12
iii. Repeat steps i and ii until no more
quotient may be entered. Then we may
put the fraction N/D in the following
mixed form:
–)
Hence we may write
(2x – 6)
(x + 3)
N
D
= Q + R
D
has smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
Example E. Divide using long division(2x – 6)
(x + 3)

71. Multiplication and Division of Rational Expressions
)x + 3 2x – 6
2
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
2x + 6
–12
iii. Repeat steps i and ii until no more
quotient may be entered. Then we may
put the fraction N/D in the following
mixed form:
–)
Hence we may write
(2x – 6)
(x + 3)
N
D
= Q + R
D
has smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
Q
R
Example E. Divide using long division(2x – 6)
(x + 3)

72. Multiplication and Division of Rational Expressions
)x + 3 2x – 6
2
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
2x + 6
–12
iii. Repeat steps i and ii until no more
quotient may be entered. Then we may
put the fraction N/D in the following
mixed form:
= 2 – 12
x + 3
–)
Hence we may write
(2x – 6)
(x + 3)
N
D
= Q + R
D
has smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
Q
R
Q R
Example E. Divide using long division(2x – 6)
(x + 3)

73. Multiplication and Division of Rational Expressions
Example F. Divide using long division
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
iii. Repeat steps i and ii until no more
quotient may be entered. Then we may
put the fraction N/D in the following
mixed form:
x2 – 6x + 3
x – 2
N
D
= Q + R
D
has smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.

74. Multiplication and Division of Rational Expressions
)x + 3
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
iii. Repeat steps i and ii until no more
quotient may be entered. Then we may
put the fraction N/D in the following
mixed form:
x2 – 6x + 3
Make sure the terms
are in order.
N
D
= Q + R
D
has smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
Example F. Divide using long divisionx2 – 6x + 3
x – 2

75. Multiplication and Division of Rational Expressions
)x + 3
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
iii. Repeat steps i and ii until no more
quotient may be entered. Then we may
put the fraction N/D in the following
mixed form:
x2 – 6x + 3
N
D
= Q + R
D
has smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
Example F. Divide using long divisionx2 – 6x + 3
x – 2

76. Multiplication and Division of Rational Expressions
)x + 3
x
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
iii. Repeat steps i and ii until no more
quotient may be entered. Then we may
put the fraction N/D in the following
mixed form:
x2 – 6x + 3
N
D
= Q + R
D
has smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
Example F. Divide using long divisionx2 – 6x + 3
x – 2

77. Multiplication and Division of Rational Expressions
)x + 3
x
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
x2 + 3x
iii. Repeat steps i and ii until no more
quotient may be entered. Then we may
put the fraction N/D in the following
mixed form:
x2 – 6x + 3
N
D
= Q + R
D
has smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
Example F. Divide using long divisionx2 – 6x + 3
x – 2

78. Multiplication and Division of Rational Expressions
)x + 3
x
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
x2 + 3x
–9x + 3
iii. Repeat steps i and ii until no more
quotient may be entered. Then we may
put the fraction N/D in the following
mixed form:
–)
x2 – 6x + 3
N
D
= Q + R
D
has smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
Example F. Divide using long divisionx2 – 6x + 3
x – 2

79. Multiplication and Division of Rational Expressions
)x + 3
x – 9
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
x2 + 3x
–9x + 3
iii. Repeat steps i and ii until no more
quotient may be entered. Then we may
put the fraction N/D in the following
mixed form:
–)
x2 – 6x + 3
N
D
= Q + R
D
has smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
Example F. Divide using long divisionx2 – 6x + 3
x – 2

80. Multiplication and Division of Rational Expressions
)x + 3
x – 9
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
x2 + 3x
–9x + 3
iii. Repeat steps i and ii until no more
quotient may be entered. Then we may
put the fraction N/D in the following
mixed form:
–)
x2 – 6x + 3
–9x – 27
N
D
= Q + R
D
has smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
Example F. Divide using long divisionx2 – 6x + 3
x – 2

81. Multiplication and Division of Rational Expressions
)x + 3
x – 9
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
x2 + 3x
–9x + 3
iii. Repeat steps i and ii until no more
quotient may be entered. Then we may
put the fraction N/D in the following
mixed form:
–)
x2 – 6x + 3
–9x – 27–)
30
N
D
= Q + R
D
has smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
Example F. Divide using long divisionx2 – 6x + 3
x – 2

82. Multiplication and Division of Rational Expressions
)x + 3
x – 9
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
x2 + 3x
–9x + 3
iii. Repeat steps i and ii until no more
quotient may be entered. Then we may
put the fraction N/D in the following
mixed form:
N
D
= Q + R
D
has smaller degree then denominator D (no more quotient).
where Q is the quotient
–)
x2 – 6x + 3
–9x – 27–)
30
Stop. No more quotient
since x can’t going into
30. Hence 30 is the
remainder.
and the remainder R
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
Example F. Divide using long divisionx2 – 6x + 3
x – 2

83. Multiplication and Division of Rational Expressions
)x + 3
x – 9
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
x2 + 3x
–9x + 3
iii. Repeat steps i and ii until no more
quotient may be entered. Then we may
put the fraction N/D in the following
mixed form:
–)
x2 – 6x + 3
–9x – 27–)
30
Hence
x2 – 6x + 3
x – 2
=
N
D
= Q + R
D
has smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
Example F. Divide using long divisionx2 – 6x + 3
x – 2

84. Multiplication and Division of Rational Expressions
)x + 3
x – 9
ii. Multiply the quotient back into the
problem and subtract the results from
the numerator. Bring down the
remaining terms from the numerator.
x2 + 3x
–9x + 3
iii. Repeat steps i and ii until no more
quotient may be entered. Then we may
put the fraction N/D in the following
mixed form:
–)
x2 – 6x + 3
–9x – 27–)
30
Hence
x2 – 6x + 3
x – 2
= x – 9 + 30
x + 3
i. Put the problem in the long division
format with the “bottom-out” and
enter the quotients of the leading
terms.
N
D
= Q + R
D
has smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
Example F. Divide using long divisionx2 – 6x + 3
x – 2

85. Ex A. Simplify. Do not expand the results.
Multiplication and Division of Rational Expressions
1. 10x *
2
5x3
15x
4
*
16
25x4
10x
*
35x32.
5. 10
9x4
*
18
5x3
6.
3.12x6*
5
6x14
56x6
27
*
63
8x5
10x
*
35x34.
7. 75x
49
*
42
25x3
8.
9.
2x – 4
2x + 4
5x + 10
3x – 6
10.
6 – 4x
3x – 2
x – 2
2x + 4
11.
9x – 12
2x – 4
2 – x
8 – 6x
12.
x + 4
–x – 4
4 – x
x – 4
13.
3x – 9
15x – 5
3 – x
5 – 15x
14.
42 – 6x
–2x + 14
4 – 2x
–7x + 14
*
*
*
*
*
*
15.
(x2 + x – 2 )
(x – 2) (x2 – x)
(x2 – 4 )
*
16.
(x2 + 2x – 3 )
(x2 – 9) (x2 – x – 2 )
(x2 – 2x – 3)
*
17.
(x2 – x – 2 )
(x2 – 1) (x2 + 2x + 1)
(x2 + x )
*
18.
(x2 + 5x – 6 )
(x2 + 5x + 6) (x2 – 5x – 6 )
(x2 – 5x + 6)
*
19.
(x2 – 3x – 4 )
(x2 – 1) (x2 – 2x – 8)
(x2 – 3x + 2)
*
20.
(– x2 + 6 – x )
(x2 + 5x + 6) (x2 – x – 12 )
(6 – x2 – x)
*

86. Ex. A. Simplify. Do not expand the results.
Multiplication and Division of Rational Expressions
21.
(2x2 + x – 1 )
(1 – 2x)
(4x2 – 1)
(2x2 – x )
22.
(3x2 – 2x – 1)
(1 – 9x2)
(x2 + x – 2 )
(x2 + 4x + 4)
23.(3x2 – x – 2)
(x2 – x + 2) (3x2 + 4x + 1)
(–x – 3x2)
24.
(x + 1 – 6x2)
(–x2 – 4)
(2x2 + x – 1 )
(x2 – 5x – 6)
25. (x3 – 4x)
(–x2 + 4x – 4)
(x2 + 2)
(–x + 2)
26.
(–x3 + 9x ) (x2 + 6x + 9)
(x2 + 3x) (–3x2 – 9x)
Ex. B. Multiply, expand and simplify the results.
÷
÷
÷
÷
÷
÷
27. x + 3
x + 1
(x2 – 1) 28. x – 3
x – 2
(x2 – 4) 29. 2x + 3
1 – x
(x2 – 1)
30. 3 – 2x
x + 2 (x + 2)(x +1) 31. 3 – 2x
2x – 1 (3x + 2)(1 – 2x)
32. x – 2
x – 3
( +
x + 1
x + 3
)( x – 3)(x + 3)
33. 2x – 1
x + 2
( –
x + 2
2x – 3
) ( 2x – 3)(x + 2)

87. Multiplication and Division of Rational Expressions
38. x – 2
x2 – 9
( –
x + 1
x2 – 2x – 3
) ( x – 3)(x + 3)(x + 1)
39. x + 3
x2 – 4
( –
2x + 1
x2 + x – 2
) ( x – 2)(x + 2)(x – 1)
40. x – 1
x2 – x – 6
( –
x + 1
x2 – 2x – 3
) ( x – 3)(x + 2)(x + 1)
41. x + 2
x2 – 4x +3
( –
2x + 1
x2 + 2x – 3
)( x – 3)(x + 3)(x – 1)
34. 4 – x
x – 3
( –
x – 1
2x + 3
)( x – 3)(2x + 3)
35. 3 – x
x + 2
( – 2x + 3
x – 3
)(x – 3)(x + 2)
Ex B. Multiply, expand and simplify the results.
36. 3 – 4x
x + 1
( –
1 – 2x
x + 3
)( x + 3)(x + 1)
37. 5x – 7
x + 5
( –
4 – 5x
x – 3
)(x – 3)(x + 5)

88. Ex. C. Break up the following expressions as sums and
differences of fractions.
42.
43. 44.
45. 46. 47.
x2 + 4x – 6
2x2x2 – 4
x2
12x3 – 9x2 + 6x
3x
x2 – 4
2x
x
x8 – x6 – x4
x2
x8 – x6 – x4
Ex D. Use long division and write each rational expression in
the form of Q + .
R
D
(x2 + x – 2 )
(x – 1)
(3x2 – 3x – 2 )
(x + 2)
2x + 6
x + 2
48.
3x – 5
x – 2
49.
4x + 3
x – 1
50.
5x – 4
x – 3
51.
3x + 8
2 – x
52.
–4x – 5
1 – x
53.
54. (2x2 + x – 3 )
(x – 2)
55. 56.
(–x2 + 4x – 3 )
(x – 3)
(5x2 – 1 )
(x – 4)
57. (4x2 + 2 )
(x + 3)
58. 59.
Multiplication and Division of Rational Expressions