The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions, which states that the product of two rational expressions is equal to the product of the tops divided by the product of the bottoms. Examples are provided to demonstrate simplifying rational expression products by factoring and canceling like terms.
Marketing measurement used to be all about impressions, page views, open and click-through rates, but those days are gone.
Ultimately, the metrics that matter the most are the ones directly related to achieving revenue targets -- the ones that tie to growth, market acceptance, and sales. When you can demonstrate the correlation, you can justify your marketing spend, and the conversation will change.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
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.
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
