Rational expressions are expressions of the form P/Q, where P and Q are polynomials. Polynomials are expressions of the form anxn + an-1xn-1 + ... + a1x1 + a0. Rational expressions can be written in either expanded or factored form. The factored form is useful for determining the domain of a rational expression, solving equations involving rational expressions, evaluating inputs, and determining the sign of outputs. The domain of a rational expression excludes values of x that make the denominator equal to 0.
GR 8 Math Powerpoint about Polynomial Techniquesreginaatin
-This is a powerpoint inspired by one of Canva displayed presentation.
- This is about Math Polynomials and good for highschoolers presentation for school.
- It consists of 39 pages explaining each of the Polynomial Techniques.
- Good for review or inspired powerpoint.
This is a PowerPoint on teaching the subject of Polynomials, Monomials, and Rational Expressions, dealing with how to add, subtract, multiply, and simplify all of these.
GR 8 Math Powerpoint about Polynomial Techniquesreginaatin
-This is a powerpoint inspired by one of Canva displayed presentation.
- This is about Math Polynomials and good for highschoolers presentation for school.
- It consists of 39 pages explaining each of the Polynomial Techniques.
- Good for review or inspired powerpoint.
This is a PowerPoint on teaching the subject of Polynomials, Monomials, and Rational Expressions, dealing with how to add, subtract, multiply, and simplify all of these.
Rational Expressions
GRADE 8 - MATHEMATICS
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Rational Expressions
GRADE 8 - MATHEMATICS
To get/buy a soft copy, please send a request to queenyedda@gmail.com
Inclusions of the file attachment:
* Fonts used
* Soft copy of the whole ppt file with effects
* Complete activities
PRICE: P200 only
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
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
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
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.
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.
3. Rational Expressions
For example,
7x12 – 4x5 + 6x2 – 5,
Polynomials are expressions of the form
anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0
where the a’s are numbers.
4. Rational Expressions
For example,
7x12 – 4x5 + 6x2 – 5, πx – 0.762 are polynomials.
Polynomials are expressions of the form
anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0
where the a’s are numbers.
5. Rational Expressions
For example,
7x12 – 4x5 + 6x2 – 5, πx – 0.762 are polynomials.
Rational (fractional) expressions are expressions of the
form , where P and Q are polynomials.P
Q
Polynomials are expressions of the form
anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0
where the a’s are numbers.
6. Rational Expressions
For example,
7x12 – 4x5 + 6x2 – 5, πx – 0.762 are polynomials.
Rational (fractional) expressions are expressions of the
form , where P and Q are polynomials.P
Q
Polynomials are expressions of the form
anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0
where the a’s are numbers.
All polynomials are rational expressions by viewing P as .
P
1
7. Rational Expressions
For example,
7x12 – 4x5 + 6x2 – 5, πx – 0.762 are polynomials.
Rational (fractional) expressions are expressions of the
form , where P and Q are polynomials.P
Q
Polynomials are expressions of the form
anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0
where the a’s are numbers.
All polynomials are rational expressions by viewing P as .
P
1
x – 2
x2 – 2 x + 1 ,
8. Rational Expressions
For example,
7x12 – 4x5 + 6x2 – 5, πx – 0.762 are polynomials.
Rational (fractional) expressions are expressions of the
form , where P and Q are polynomials.P
Q
Polynomials are expressions of the form
anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0
where the a’s are numbers.
All polynomials are rational expressions by viewing P as .
P
1
x – 2
x2 – 2 x + 1 ,
x(x – 2)
(x + 1) (2x + 1) are rational expressions.
9. Rational Expressions
For example,
7x12 – 4x5 + 6x2 – 5, πx – 0.762 are polynomials.
Rational (fractional) expressions are expressions of the
form , where P and Q are polynomials.P
Q
Polynomials are expressions of the form
anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0
where the a’s are numbers.
All polynomials are rational expressions by viewing P as .
P
1
x – 2
x2 – 2 x + 1 ,
x(x – 2)
(x + 1) (2x + 1) are rational expressions.
x – 2
2 x + 1
is not a rational expression because the
denominator is not a polynomial.
10. Rational Expressions
For example,
7x12 – 4x5 + 6x2 – 5, πx – 0.762 are polynomials.
Rational (fractional) expressions are expressions of the
form , where P and Q are polynomials.P
Q
Polynomials are expressions of the form
anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0
where the a’s are numbers.
All polynomials are rational expressions by viewing P as .
P
1
x – 2
x2 – 2 x + 1 ,
x(x – 2)
(x + 1) (2x + 1) are rational expressions.
x – 2
2 x + 1
is not a rational expression because the
denominator is not a polynomial.
Rational expressions are expressions that describe
calculation procedures that involve division (of polynomials).
11. Rational Expressions
Just as polynomials may be given in the expanded form or
factored form, rational expressions may also be given in
these two forms.
12. Rational Expressions
Just as polynomials may be given in the expanded form or
factored form, rational expressions may also be given in
these two forms.
The rational expression
x2 – 4
x2 + 2x + 1
is in the expanded form.
13. Rational Expressions
Just as polynomials may be given in the expanded form or
factored form, rational expressions may also be given in
these two forms.
The rational expression
x2 – 4
(x + 2)(x – 2)
(x + 1)(x + 1) .
is in the expanded form.
In the factored form, it’s
x2 + 2x + 1
14. Rational Expressions
Just as polynomials may be given in the expanded form or
factored form, rational expressions may also be given in
these two forms.
The rational expression
x2 – 4
x2 + 2x + 1
(x + 2)(x – 2)
(x + 1)(x + 1) .
x – 2
x2 + 1The expression is in both forms.
is in the expanded form.
In the factored form, it’s
Example A. Put the following expressions in the factored form.
a. x2 – 3x – 10
x2 – 3x
b. x2 – 3x + 10
x2 – 3
15. Rational Expressions
Just as polynomials may be given in the expanded form or
factored form, rational expressions may also be given in
these two forms.
The rational expression
x2 – 4
x2 + 2x + 1
(x + 2)(x – 2)
(x + 1)(x + 1) .
x – 2
x2 + 1The expression is in both forms.
is in the expanded form.
Example A. Put the following expressions in the factored form.
(x – 5)(x + 2)a.
x(x – 3)
x2 – 3x – 10
x2 – 3x
=
b. x2 – 3x + 10
x2 – 3
In the factored form, it’s
16. Rational Expressions
Just as polynomials may be given in the expanded form or
factored form, rational expressions may also be given in
these two forms.
The rational expression
x2 – 4
x2 + 2x + 1
(x + 2)(x – 2)
(x + 1)(x + 1) .
x – 2
x2 + 1The expression is in both forms.
is in the expanded form.
Example A. Put the following expressions in the factored form.
(x – 5)(x + 2)a.
x(x – 3)
x2 – 3x – 10
x2 – 3x
=
b. x2 – 3x + 10
x2 – 3
is in the factored form
In the factored form, it’s
17. Rational Expressions
Just as polynomials may be given in the expanded form or
factored form, rational expressions may also be given in
these two forms.
The rational expression
x2 – 4
x2 + 2x + 1
(x + 2)(x – 2)
(x + 1)(x + 1) .
x – 2
x2 + 1The expression is in both forms.
is in the expanded form.
Example A. Put the following expressions in the factored form.
(x – 5)(x + 2)a.
x(x – 3)
x2 – 3x – 10
x2 – 3x
=
b. x2 – 3x + 10
x2 – 3
is in the factored form
Note that in b. the entire (x2 – 3x + 10) or (x2 – 3) are viewed
as a single factors because they can’t be factored further.
In the factored form, it’s
18. We use the factored form to
1. solve equations
Rational Expressions
19. We use the factored form to
1. solve equations
2. determine the domain of rational expressions
Rational Expressions
20. We use the factored form to
1. solve equations
2. determine the domain of rational expressions
2. evaluate given inputs
Rational Expressions
21. We use the factored form to
1. solve equations
2. determine the domain of rational expressions
2. evaluate given inputs
3. determine of the signs of the outputs
Rational Expressions
22. We use the factored form to
1. solve equations
2. determine the domain of rational expressions
2. evaluate given inputs
3. determine of the signs of the outputs
Solutions of Equations
Rational Expressions
23. We use the factored form to
1. solve equations
2. determine the domain of rational expressions
2. evaluate given inputs
3. determine of the signs of the outputs
Solutions of Equations
The solutions of the equation of the form = 0 are the
zeroes of the numerator, so they are the solutions of P = 0.
Rational Expressions
P
Q
24. We use the factored form to
1. solve equations
2. determine the domain of rational expressions
2. evaluate given inputs
3. determine of the signs of the outputs
Solutions of Equations
Example B. a. Write in the factored form and
solve the equation
The solutions of the equation of the form = 0 are the
zeroes of the numerator, so they are the solutions of P = 0.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
x3 + x2 – 2x
x2 + 4x + 3
= 0
P
Q
25. We use the factored form to
1. solve equations
2. determine the domain of rational expressions
2. evaluate given inputs
3. determine of the signs of the outputs
Solutions of Equations
Example B. a. Write in the factored form and
solve the equation
The solutions of the equation of the form = 0 are the
zeroes of the numerator, so they are the solutions of P = 0.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
x3 – 2x2 + 3x
x2 + 4x + 3
x3 + x2 – 2x
x2 + 4x + 3
= 0
Factor, we get
P
Q
=
26. We use the factored form to
1. solve equations
2. determine the domain of rational expressions
2. evaluate given inputs
3. determine of the signs of the outputs
Solutions of Equations
Example B. a. Write in the factored form and
solve the equation
The solutions of the equation of the form = 0 are the
zeroes of the numerator, so they are the solutions of P = 0.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
x3 – 2x2 + 3x
x2 + 4x + 3
= x(x + 2)(x – 1)
(x + 3)(x + 1)
x3 + x2 – 2x
x2 + 4x + 3
= 0
Factor, we get
P
Q
27. We use the factored form to
1. solve equations
2. determine the domain of rational expressions
2. evaluate given inputs
3. determine of the signs of the outputs
Solutions of Equations
Example B. a. Write in the factored form and
solve the equation
The solutions of the equation of the form = 0 are the
zeroes of the numerator, so they are the solutions of P = 0.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
x3 – 2x2 + 3x
x2 + 4x + 3
= x(x + 2)(x – 1)
(x + 3)(x + 1)
x3 + x2 – 2x
x2 + 4x + 3
= 0
Factor, we get
Hence for
x3 + x2 – 2x
x2 + 4x + 3
= 0, it must be that x(x + 2)(x – 1) = 0
P
Q
28. We use the factored form to
1. solve equations
2. determine the domain of rational expressions
2. evaluate given inputs
3. determine of the signs of the outputs
Solutions of Equations
Example B. a. Write in the factored form and
solve the equation
The solutions of the equation of the form = 0 are the
zeroes of the numerator, so they are the solutions of P = 0.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
x3 – 2x2 + 3x
x2 + 4x + 3
= x(x + 2)(x – 1)
(x + 3)(x + 1)
x3 + x2 – 2x
x2 + 4x + 3
= 0
Factor, we get
Hence for
x3 + x2 – 2x
x2 + 4x + 3
= 0, it must be that x(x + 2)(x – 1) = 0
or that x = 0, –2, 1.
P
Q
29. Domain
The domain of a formula is the set of all the numbers that we
may use as input values for x.
Rational Expressions
30. Domain
The domain of a formula is the set of all the numbers that we
may use as input values for x. Because the denominator of a
fraction can’t be 0, therefore for the rational formulas
the zeroes of the denominator Q can’t be used as inputs.
Rational Expressions
P
Q
31. Domain
The domain of a formula is the set of all the numbers that we
may use as input values for x. Because the denominator of a
fraction can’t be 0, therefore for the rational formulas
the zeroes of the denominator Q can’t be used as inputs.
in other words, the domain of are all the numbers except
where Q = 0.
Rational Expressions
P
Q
P
Q
32. Domain
b. Determine the domain of the formula
The domain of a formula is the set of all the numbers that we
may use as input values for x. Because the denominator of a
fraction can’t be 0, therefore for the rational formulas
the zeroes of the denominator Q can’t be used as inputs.
in other words, the domain of are all the numbers except
where Q = 0.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
P
Q
P
Q
33. Domain
b. Determine the domain of the formula
The domain of a formula is the set of all the numbers that we
may use as input values for x. Because the denominator of a
fraction can’t be 0, therefore for the rational formulas
the zeroes of the denominator Q can’t be used as inputs.
in other words, the domain of are all the numbers except
where Q = 0.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
x3 – 2x2 + 3x
x2 + 4x + 3
Factor expression first.
= x(x + 2)(x – 1)
(x + 3)(x + 1)
P
Q
P
Q
34. Domain
b. Determine the domain of the formula
The domain of a formula is the set of all the numbers that we
may use as input values for x. Because the denominator of a
fraction can’t be 0, therefore for the rational formulas
the zeroes of the denominator Q can’t be used as inputs.
in other words, the domain of are all the numbers except
where Q = 0.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
x3 – 2x2 + 3x
x2 + 4x + 3
Factor expression first.
= x(x + 2)(x – 1)
(x + 3)(x + 1)
Hence we can’t have
P
Q
P
Q
(x + 3)(x + 1) = 0
35. Domain
b. Determine the domain of the formula
The domain of a formula is the set of all the numbers that we
may use as input values for x. Because the denominator of a
fraction can’t be 0, therefore for the rational formulas
the zeroes of the denominator Q can’t be used as inputs.
in other words, the domain of are all the numbers except
where Q = 0.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
x3 – 2x2 + 3x
x2 + 4x + 3
Factor expression first.
= x(x + 2)(x – 1)
(x + 3)(x + 1)
Hence we can’t have
P
Q
P
Q
(x + 3)(x + 1) = 0
so that the domain is the set of all the numbers except
–1 and –3.
36. Evaluation
It is often easier to evaluate expressions in the factored form.
Rational Expressions
37. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
38. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
Using the factored form
x(x + 2)(x – 1)
(x + 3)(x + 1)
Plug in x = 7,
39. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
Using the factored form
x(x + 2)(x – 1)
(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)
(10)(8)
40. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
Using the factored form
x(x + 2)(x – 1)
(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)
(10)(8)
3
4
41. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
Using the factored form
x(x + 2)(x – 1)
(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)
(10)(8)
=
3
4
189
40
42. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
Using the factored form
x(x + 2)(x – 1)
(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)
(10)(8)
=
3
4
189
40
Signs
43. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
Using the factored form
x(x + 2)(x – 1)
(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)
(10)(8)
=
3
4
189
40
We use the factored form to determine the sign of an output.
Signs
44. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
Using the factored form
x(x + 2)(x – 1)
(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)
(10)(8)
=
3
4
189
40
We use the factored form to determine the sign of an output.
Signs
d. Determine the signs (positive or negative) of
if x = ½,– 5/2 using the factored form.
x3 + x2 – 2x
x2 + 4x + 3
45. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
Using the factored form
x(x + 2)(x – 1)
(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)
(10)(8)
=
3
4
189
40
We use the factored form to determine the sign of an output.
Signs
d. Determine the signs (positive or negative) of
if x = ½,– 5/2 using the factored form.
x3 + x2 – 2x
x2 + 4x + 3
Plug in x = ½ into the above factored form and check the signs
of each factor,
46. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
Using the factored form
x(x + 2)(x – 1)
(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)
(10)(8)
=
3
4
189
40
We use the factored form to determine the sign of an output.
Signs
d. Determine the signs (positive or negative) of
if x = ½,– 5/2 using the factored form.
x3 + x2 – 2x
x2 + 4x + 3
Plug in x = ½ into the above factored form and check the signs
of each factor, we get +( + )( – )
(+)(+)
47. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
Using the factored form
x(x + 2)(x – 1)
(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)
(10)(8)
=
3
4
189
40
We use the factored form to determine the sign of an output.
Signs
d. Determine the signs (positive or negative) of
if x = ½,– 5/2 using the factored form.
x3 + x2 – 2x
x2 + 4x + 3
Plug in x = ½ into the above factored form and check the signs
of each factor, we get = –, so it’s negative.+( + )( – )
(+)(+)
48. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
Using the factored form
x(x + 2)(x – 1)
(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)
(10)(8)
=
3
4
189
40
We use the factored form to determine the sign of an output.
Signs
d. Determine the signs (positive or negative) of
if x = ½,– 5/2 using the factored form.
x3 + x2 – 2x
x2 + 4x + 3
Plug in x = ½ into the above factored form and check the signs
of each factor, we get
Plug in x = –5/2 into the factored form we get –( – )( – )
(+)( – )
= –, so it’s negative.+( + )( – )
(+)(+)
49. Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2x
x2 + 4x + 3
Using the factored form
x(x + 2)(x – 1)
(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)
(10)(8)
=
3
4
189
40
We use the factored form to determine the sign of an output.
Signs
d. Determine the signs (positive or negative) of
if x = ½,– 5/2 using the factored form.
x3 + x2 – 2x
x2 + 4x + 3
Plug in x = ½ into the above factored form and check the signs
of each factor, we get = –, so it’s negative.+( + )( – )
(+)(+)
Plug in x = –5/2 into the factored form we get –( – )( – )
(+)( – ) = +
so that the output is positive.
51. PN
QN
= =
1
Rational Expressions
Cancellation Law: Common factor may be cancelled, i.e.
PN
QN
P
Q
A rational expression is reduced (simplified) if all common
factors are cancelled.
52. PN
QN
= =
1
Rational Expressions
Cancellation Law: Common factor may be cancelled, i.e.
PN
QN
P
Q
A rational expression is reduced (simplified) if all common
factors are cancelled.
To reduce (simplify) a rational expression, put the expression
in the factored form then cancel the common factors.
53. PN
QN
= =
1
Rational Expressions
Cancellation Law: Common factor may be cancelled, i.e.
PN
QN
P
Q
A rational expression is reduced (simplified) if all common
factors are cancelled.
To reduce (simplify) a rational expression, put the expression
in the factored form then cancel the common factors.
Example B. Reduce the following expressions.
(x – 2)(x + 3)
(x + 3)(x + 2)
a.
b.
x2 – 3x + 10
x2 – 3
54. PN
QN
= =
1
Rational Expressions
Cancellation Law: Common factor may be cancelled, i.e.
PN
QN
P
Q
A rational expression is reduced (simplified) if all common
factors are cancelled.
To reduce (simplify) a rational expression, put the expression
in the factored form then cancel the common factors.
Example B. Reduce the following expressions.
(x – 2)(x + 3)
It's already factored, proceed to cancel the common factor.
(x + 3)(x + 2)
(x – 2)(x + 3)
(x + 3)(x + 2)
a.
b.
x2 – 3x + 10
x2 – 3
55. PN
QN
= =
1
Rational Expressions
Cancellation Law: Common factor may be cancelled, i.e.
PN
QN
P
Q
A rational expression is reduced (simplified) if all common
factors are cancelled.
To reduce (simplify) a rational expression, put the expression
in the factored form then cancel the common factors.
Example B. Reduce the following expressions.
(x – 2)(x + 3)
1
=
x – 2
x + 2
It's already factored, proceed to cancel the common factor.
(x + 3)(x + 2)
(x – 2)(x + 3)
(x + 3)(x + 2)
which is reduced.
a.
b.
x2 – 3x + 10
x2 – 3
56. PN
QN
= =
1
Rational Expressions
Cancellation Law: Common factor may be cancelled, i.e.
PN
QN
P
Q
A rational expression is reduced (simplified) if all common
factors are cancelled.
To reduce (simplify) a rational expression, put the expression
in the factored form then cancel the common factors.
Example B. Reduce the following expressions.
(x – 2)(x + 3)
1
=
x – 2
x + 2
It's already factored, proceed to cancel the common factor.
(x + 3)(x + 2)
(x – 2)(x + 3)
(x + 3)(x + 2)
which is reduced.
a.
b.
x2 – 3x + 10
x2 – 3
This is in the factored form.
57. PN
QN
= =
1
Rational Expressions
Cancellation Law: Common factor may be cancelled, i.e.
PN
QN
P
Q
A rational expression is reduced (simplified) if all common
factors are cancelled.
To reduce (simplify) a rational expression, put the expression
in the factored form then cancel the common factors.
Example B. Reduce the following expressions.
(x – 2)(x + 3)
1
=
x – 2
x + 2
It's already factored, proceed to cancel the common factor.
(x + 3)(x + 2)
(x – 2)(x + 3)
(x + 3)(x + 2)
which is reduced.
a.
b.
x2 – 3x + 10
x2 – 3
This is in the factored form. There are no common factors
so it’s already reduced.
62. Rational Expressions
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
63. Rational Expressions
Only factors may be canceled.
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
64. Rational Expressions
Only factors may be canceled. It is wrong to cancel terms
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
65. Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e.
P + Q
P + R
=
P + Q
P + R
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
66. Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e.
P + Q
P + R
=
P + Q
P + R
For example,
x2 + 1
x2 – 2 =
x2 + 1
x2 – 2
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
67. Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e.
P + Q
P + R
=
P + Q
P + R
For example,
x2 + 1
x2 – 2 =
x2 + 1
x2 – 2
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
Cancellation of Opposite Factors
68. Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e.
P + Q
P + R
=
P + Q
P + R
For example,
x2 + 1
x2 – 2 =
x2 + 1
x2 – 2
The opposite of a quantity x is the –x.
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
Cancellation of Opposite Factors
69. Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e.
P + Q
P + R
=
P + Q
P + R
For example,
x2 + 1
x2 – 2 =
x2 + 1
x2 – 2
The opposite of a quantity x is the –x.
original: y -z x – y v – 4u – 2w
opposite:
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
Cancellation of Opposite Factors
70. Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e.
P + Q
P + R
=
P + Q
P + R
For example,
x2 + 1
x2 – 2 =
x2 + 1
x2 – 2
The opposite of a quantity x is the –x.
original: y -z x – y v – 4u – 2w
opposite: -y
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
Cancellation of Opposite Factors
71. Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e.
P + Q
P + R
=
P + Q
P + R
For example,
x2 + 1
x2 – 2 =
x2 + 1
x2 – 2
The opposite of a quantity x is the –x.
original: y -z x – y v – 4u – 2w
opposite: -y z
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
Cancellation of Opposite Factors
72. Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e.
P + Q
P + R
=
P + Q
P + R
For example,
x2 + 1
x2 – 2 =
x2 + 1
x2 – 2
The opposite of a quantity x is the –x.
original: y -z x – y v – 4u – 2w
opposite: -y z -x + y or y – x
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
Cancellation of Opposite Factors
73. Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e.
P + Q
P + R
=
P + Q
P + R
For example,
x2 + 1
x2 – 2 =
x2 + 1
x2 – 2
The opposite of a quantity x is the –x.
original: y -z x – y v – 4u – 2w
opposite: -y z -x + y or y – x -v + 4u + 2w or …
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
Cancellation of Opposite Factors
74. Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e.
P + Q
P + R
=
P + Q
P + R
For example,
x2 + 1
x2 – 2 =
x2 + 1
x2 – 2
Cancellation of Opposite Factors
The opposite of a quantity x is the –x.
While identical factors cancel to be 1, opposite factors
cancel to be –1,
original: y -z x – y v – 4u – 2w
opposite: -y z -x + y or y – x -v + 4u + 2w or …
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
75. Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e.
P + Q
P + R
=
P + Q
P + R
For example,
x2 + 1
x2 – 2 =
x2 + 1
x2 – 2
The opposite of a quantity x is the –x.
While identical factors cancel to be 1, opposite factors
cancel to be –1, in symbol,
original: y -z x – y v – 4u – 2w
opposite: -y z -x + y or y – x -v + 4u + 2w or …
x
–x = –1.
c.
x2 – 1
x2 – 3x+ 2
x2 – 1
x2 – 3x+ 2
=
(x – 1)(x + 1)
(x – 1)(x – 2)
1
=
x + 1
x – 2
Factor then cancel
Cancellation of Opposite Factors
78. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1a.
b.
Rational Expressions
79. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
a.
b.
Rational Expressions
80. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
81. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
82. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
-1
83. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
-1
=
–(x + 3y)
(x – 3y)
84. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
-1
=
–(x + 3y)
(x – 3y)
or
–x – 3y
x – 3y
85. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
-1
=
–(x + 3y)
(x – 3y)
or
–x – 3y
x – 3y
d.
4 – x2
x2 – x – 2
86. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
-1
=
–(x + 3y)
(x – 3y)
or
–x – 3y
x – 3y
d.
4 – x2
x2 – x – 2
=
(2 – x)(2 + x)
(x + 1)(x – 2)
87. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
-1
=
–(x + 3y)
(x – 3y)
or
–x – 3y
x – 3y
d.
4 – x2
x2 – x – 2
=
(2 – x)(2 + x)
(x + 1)(x – 2)
-1
88. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
-1
=
–(x + 3y)
(x – 3y)
or
–x – 3y
x – 3y
d.
4 – x2
x2 – x – 2
=
(2 – x)(2 + x)
(x + 1)(x – 2)
-1
=
–(2 + x)
x + 1
or
–2 – x
x + 1
89. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
-1
=
–(x + 3y)
(x – 3y)
or
–x – 3y
x – 3y
d.
4 – x2
x2 – x – 2
=
(2 – x)(2 + x)
(x + 1)(x – 2)
-1
=
–(2 + x)
x + 1
or
–2 – x
x + 1
In the case of polynomials in one variable x, if the highest
degree term is negative, we may factor out the negative sign
then factor the expressions.
90. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
-1
=
–(x + 3y)
(x – 3y)
or
–x – 3y
x – 3y
d.
4 – x2
x2 – x – 2
=
(2 – x)(2 + x)
(x + 1)(x – 2)
-1
=
–(2 + x)
x + 1
or
–2 – x
x + 1
–x2 + 4
–x2 + x + 2
Example D. Pull out the “–” first then reduce.
In the case of polynomials in one variable x, if the highest
degree term is negative, we may factor out the negative sign
then factor the expressions.
91. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
-1
=
–(x + 3y)
(x – 3y)
or
–x – 3y
x – 3y
d.
4 – x2
x2 – x – 2
=
(2 – x)(2 + x)
(x + 1)(x – 2)
-1
=
–(2 + x)
x + 1
or
–2 – x
x + 1
In the case of polynomials in one variable x, if the highest
degree term is negative, we may factor out the negative sign
then factor the expressions.
–x2 + 4
–x2 + x + 2
=
Example D. Pull out the “–” first then reduce.
–(x2 – 4)
–(x2 – x – 2)
92. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
-1
=
–(x + 3y)
(x – 3y)
or
–x – 3y
x – 3y
d.
4 – x2
x2 – x – 2
=
(2 – x)(2 + x)
(x + 1)(x – 2)
-1
=
–(2 + x)
x + 1
or
–2 – x
x + 1
–x2 + 4
–x2 + x + 2
=
(x – 2)(x + 2)
(x + 1)(x – 2)
=
Example D. Pull out the “–” first then reduce.
–(x2 – 4)
–(x2 – x – 2)
In the case of polynomials in one variable x, if the highest
degree term is negative, we may factor out the negative sign
then factor the expressions.
93. b(x – y)
a(y – x)
Example C.
2y
–2y
=
-1
–1
-1
=
–b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y)
(x – 3y)(2y – x)
c.
-1
=
–(x + 3y)
(x – 3y)
or
–x – 3y
x – 3y
d.
4 – x2
x2 – x – 2
=
(2 – x)(2 + x)
(x + 1)(x – 2)
-1
=
–(2 + x)
x + 1
or
–2 – x
x + 1
–x2 + 4
–x2 + x + 2
=
(x – 2)(x + 2)
(x + 1)(x – 2)
=
Example D. Pull out the “–” first then reduce.
–(x2 – 4)
–(x2 – x – 2)
=
x + 2
x + 1
In the case of polynomials in one variable x, if the highest
degree term is negative, we may factor out the negative sign
then factor the expressions.
94. Rational Expressions
To summarize, a rational expression is reduced (simplified)
if all common factors are cancelled.
Following are the steps for reducing a rational expression.
1. Factor the top and bottom completely.
(If present, factor the “ – ” from the leading term)
1. Cancel the common factors:
-cancel identical factors to be 1
-cancel opposite factors to be –1
95. Ex. A. Write the following expressions in factored form.
List all the distinct factors of the numerator and the
denominator of each expression.
1.
Rational Expressions
2x + 3
x + 3
2.
4x + 6
2x + 6
3.
x2 – 4
2x + 4
4.
x2 + 4
x2 + 4x
5.
x2 – 2x – 3
x2 + 4x
6.
x3 – 2x2 – 8x
x2 + 2x – 3
7. Find the zeroes and list the domain of
x2 – 2x – 3
x2 + 4x
8. Use the factored form to evaluate x2 – 2x – 3
x2 + 4x
with x = 7, ½, – ½, 1/3.
9. Determine the signs of the outputs of
x2 – 2x – 3
x2 + 4x
with x = 4, –2, 1/7, 1.23.
For problems 10, 11, and 12, answer the same questions
as problems 7, 8 and 9 with the formula .x3 – 2x2 – 8x
x2 + 2x – 3
96. Ex. B. Reduce the following expressions. If it’s already
reduced, state this. Make sure you do not cancel any terms
and make sure that you look for the opposite cancellation.
13.
Rational Expressions
2x + 3
x + 3
20.
4x + 6
2x + 3
22. 23. 24.
21.
3x – 12
x – 4
12 – 3x
x – 4
4x + 6
–2x – 3
3x + 12
x – 4
25. 4x – 6
–2x – 3
14.
x + 3
x – 3
15.
x + 3
–x – 3
16.
x + 3
x – 3
17.
x – 3
3 – x
18.
2x – 1
1 + 2x
19. 2x – 1
1 – 2x
26. (2x – y)(x – 2y)
(2y + x)(y – 2x)
27. (3y + x)(3x –y)
(y – 3x)(–x – 3y)
28. (2u + v – w)(2v – u – 2w)
(u – 2v + 2w)(–2u – v – w)
29.(a + 4b – c)(a – b – c)
(c – a – 4b)(a + b + c)
97. 30.
Rational Expressions
37.
x2 – 1
x2 + 2x – 3
36. 38.
x – x2
39.
x2 – 3x – 4
31. 32.
33. 34. 35.
40. 41. x3 – 16x
x2 + 4
2x + 4
x2 – 4x + 4
x2 – 4
x2– 2x
x2 – 9
x2 + 4x + 3
x2 – 4
2x + 4
x2 + 3x + 2
x2 – x – 2 x2 + x – 2
x2 – x – 6
x2 – 5x + 6
x2 – x – 2
x2 + x – 2
x2 – 5x – 6
x2 + 5x – 6
x2 + 5x + 6
x3 – 8x2 – 20x
46.45. 47. 9 – x2
42. 43. 44.
x2 – 2x
9 – x2
x2 + 4x + 3
– x2 – x + 2
x3 – x2 – 6x
–1 + x2
–x2 + x + 2
x2 – x – 2
– x2 + 5x – 6
1 – x2
x2 + 5x – 6
49.48. 50.
xy – 2y + x2 – 2x
x2 – y2x3 – 100x
x2 – 4xy + x – 4y
x2 – 3xy – 4y2
Ex. C. Reduce the following expressions. If it’s already
reduced, state this. Make sure you do not cancel any terms
and make sure that you look for the opposite cancellation.