13 multiplication and division of rational expressions

Multiplication and Division of Rational Expressions

Multiplication Rule for Rational Expressions A B

C D

* =

AC BD



C D

* =

AC BD


In most problems, we reduce the product by factoring the top and the bottom, then cancel.


C D

* =

AC BD



Example A. Simplify

10xy3z

a. * y2

5x3


C D

* =

AC BD



Example A. Simplify

10xy3z

a. * y2

5x3 =

10xy2

5x3y3z


C D

* =

AC BD



Example A. Simplify

10xy3z

a. * y2

5x3 =

10xy2

5x3y3z =

2x2yz


C D

* =

AC BD



Example A. Simplify

10xy3z

a. * y2

5x3 =

10xy2

5x3y3z =

2x2yz

b. (x2 + 2x – 3 ) (x – 2) (x2 – x )

(x2 – 4 )*


C D

* =

AC BD



Example A. Simplify

10xy3z

a. * y2

5x3 =

10xy2

5x3y3z =

2x2yz

b. (x2 + 2x – 3 ) (x – 2) (x2 – x )

(x2 – 4 )*

=

(x + 3)(x – 1 )


C D

* =

AC BD



Example A. Simplify

10xy3z

a. * y2

5x3 =

10xy2

5x3y3z =

2x2yz

b. (x2 + 2x – 3 ) (x – 2) (x2 – x )

(x2 – 4 )*

=

(x + 3)(x – 1 ) (x + 2 )(x – 2)


C D

* =

AC BD



Example A. Simplify

10xy3z

a. * y2

5x3 =

10xy2

5x3y3z =

2x2yz

b. (x2 + 2x – 3 ) (x – 2) (x2 – x )

(x2 – 4 )*

=

(x + 3)(x – 1 ) (x – 2)

(x + 2 )(x – 2)


C D

* =

AC BD



Example A. Simplify

10xy3z

a. * y2

5x3 =

10xy2

5x3y3z =

2x2yz

b. (x2 + 2x – 3 ) (x – 2) (x2 – x )

(x2 – 4 )*

=

(x + 3)(x – 1 ) (x – 2) x(x – 1 )

(x + 2 )(x – 2)


C D

* =

AC BD



Example A. Simplify

10xy3z

a. * y2

5x3 =

10xy2

5x3y3z =

2x2yz

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.

Multiplication and Division of Rational ExpressionsExample B. Simplify and expand the answers.

a. x + 3 x – 1 (x2 – 1)


a. x + 3 x – 1 (x2 – 1)

=

x + 3 (x – 1) (x – 1)(x + 1)

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)


a. x + 3 x – 1 (x2 – 1)

=

x + 3 (x – 1) (x – 1)(x + 1)

=

(x + 3)(x + 1) = x2 + 4x + 3



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)



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)



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)



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)



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)



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)



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)



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)



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



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

Division Rule for Rational Expressions


AB

CD

÷



AB

CD

÷ = ADBC

Reciprocate



AB

CD

÷ = ADBC

Reciprocate

We convert division by an expression of multiplying by its reciprocal.



AB

CD

÷ = ADBC

Reciprocate

We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.



AB

CD

÷ = ADBC

Reciprocate

(2x – 6) (x + 3)

÷(x2 + 2x – 3)

(9 – x2)

Example C. Simplify




AB

CD

÷ = ADBC

Reciprocate

(2x – 6) (x + 3)

÷(x2 + 2x – 3)

(9 – x2)

=(2x – 6) (x + 3)

(x2 + 2x – 3) (9 – x2)

*

Example C. Simplify




AB

CD

÷ = ADBC

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




AB

CD

÷ = ADBC

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




AB

CD

÷ = ADBC

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)




AB

CD

÷ = ADBC

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




AB

CD

÷ = ADBC

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


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

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

Multiplication and Division of Rational ExpressionsBesides 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.


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. =



(2x – 6) (x + 3)

a. =


(x + 3) – (x + 3)2x 6

(2x – 6) 3x2

b. =



(2x – 6) (x + 3)

a. =


(x + 3) – (x + 3)2x 6

(2x – 6) 3x2

b. = – 2x 63x2 3x2



(2x – 6) (x + 3)

a. =


(x + 3) – (x + 3)2x 6

(2x – 6) 3x2

b. = – 2x 63x2 3x2 = – 2

x22

3x



(2x – 6) (x + 3)

a. =


(x + 3) – (x + 3)2x 6

(2x – 6) 3x2

b. = – 2x 63x2 3x2 = – 2

x22

3xII. Long DivisionLong division is the extension of the long division of numbers from grade school and it is for the division of polynomials in one variable.



(2x – 6) (x + 3)

a. =


(x + 3) – (x + 3)2x 6

(2x – 6) 3x2

b. = – 2x 63x2 3x2 = – 2

x22

3xII. Long DivisionLong 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.

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

Multiplication and Division of Rational ExpressionsLet’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.

Multiplication and Division of Rational ExpressionsLet’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


1

ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.

845


1


845

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 move from left to right until there is enough to enter a quotient. )8 125

15


845


405

Let’s look at the example 125/8 or 125 ÷ 8 by long division.



15


845

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:ND = Q + R

Dand that R (the remainder) is smaller then D (no more quotient).

405

where Q is the quotient




15


845


Dand that R (the remainder) is smaller then D (no more quotient).

405

1258 = 15 + 5

8



Multiplication and Division of Rational ExpressionsExample E. Divide using long division(2x – 6)

(x + 3)


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)


)x + 3 2x – 6


Make sure the terms are in order.



)x + 3 2x – 6


enter the quotients of the leading terms 2x/x = 2



)x + 3 2x – 6 2


enter the quotients of the leading terms 2x/x = 2



)x + 3 2x – 6 2





)x + 3 2x – 6 2

2x + 6ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.




)x + 3 2x – 6 2


2x + 6–12

–)




)x + 3 2x – 6 2


2x + 6–12


–)




)x + 3 2x – 6 2


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.




)x + 3 2x – 6 2


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:

–)

ND = Q + R

Dhas smaller degree then denominator D (no more quotient).

where Q is the quotient and the remainder R




)x + 3 2x – 6 2


2x + 6–12


–)

Hence we may write(2x – 6) (x + 3)

ND = Q + R






)x + 3 2x – 6 2


2x + 6–12


–)


ND = Q + R




Q

R



)x + 3 2x – 6 2


2x + 6–12


= 2 – 12 x + 3

–)


ND = Q + R




Q

R

Q R


Multiplication and Division of Rational ExpressionsExample F. Divide using long division



x2 – 6x + 3x – 2

ND = Q + R





)x + 3



x2 – 6x + 3

Make sure the terms are in order.

ND = Q + R




Example F. Divide using long divisionx2 – 6x + 3x – 2


)x + 3



x2 – 6x + 3

ND = Q + R






)x + 3 x



x2 – 6x + 3

ND = Q + R






)x + 3 x


x2 + 3x


x2 – 6x + 3

ND = Q + R






)x + 3 x


x2 + 3x–9x + 3


–)x2 – 6x + 3

ND = Q + R






)x + 3 x – 9


x2 + 3x–9x + 3


–)x2 – 6x + 3

ND = Q + R






)x + 3 x – 9


x2 + 3x–9x + 3


–)x2 – 6x + 3

–9x – 27

ND = Q + R






)x + 3 x – 9


x2 + 3x–9x + 3


–)x2 – 6x + 3

–9x – 27–)30

ND = Q + R






)x + 3 x – 9


x2 + 3x–9x + 3




–)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




)x + 3 x – 9


x2 + 3x–9x + 3


–)x2 – 6x + 3

–9x – 27–)30

Hence

x2 – 6x + 3x – 2

=

ND = Q + R






)x + 3 x – 9


x2 + 3x–9x + 3


–)x2 – 6x + 3

–9x – 27–)30

Hence

x2 – 6x + 3x – 2

= x – 9 + 30x + 3


ND = Q + R




Ex A. Simplify. Do not expand the results. Multiplication and Division of Rational Expressions

1. 10x * 25x3

15x4

* 1625x4

10x* 35x32.

5. 109x4

* 185x3

6.

3.12x6* 5

6x14

56x6

27* 63

8x5

10x* 35x34.

7. 75x49

* 4225x3

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 – 44 – x

x – 4

13.3x – 9

15x – 53 – x

5 – 15x14.

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)*

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)


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)

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 + 6x3x

x2 – 4 2x

xx8 – x6 – x4

x2x8 – x6 – x4

Ex D. Use long division and write each rational expression in

the form of Q + .RD

(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 – x52. –4x – 5 1 – x53.

54. (2x2 + x – 3 ) (x – 2)

55. 56.

(–x2 + 4x – 3 ) (x – 3)

(5x2 – 1 ) (x – 4)

57. (4x2 + 2 ) (x + 3)

58. 59.


13 multiplication and division of rational expressions

Education

x2 x x2

multiplication rule

ac bd multiplication

10xy2 5x3y3z