The document discusses rational expressions, which are expressions of the form P/Q where P and Q are polynomials. Polynomials are expressions involving powers of variables with numerical coefficients. Rational expressions include polynomials as a special case where P is viewed as P/1. They may be written in expanded or factored form. The factored form is useful for determining the domain of a rational expression, solving equations involving rational expressions, evaluating expressions for given inputs, and determining the signs of outputs. The domain excludes values of x that make the denominator equal to 0.
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
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
JMeter webinar - integration with InfluxDB and GrafanaRTTS
Watch this recorded webinar about real-time monitoring of application performance. See how to integrate Apache JMeter, the open-source leader in performance testing, with InfluxDB, the open-source time-series database, and Grafana, the open-source analytics and visualization application.
In this webinar, we will review the benefits of leveraging InfluxDB and Grafana when executing load tests and demonstrate how these tools are used to visualize performance metrics.
Length: 30 minutes
Session Overview
-------------------------------------------
During this webinar, we will cover the following topics while demonstrating the integrations of JMeter, InfluxDB and Grafana:
- What out-of-the-box solutions are available for real-time monitoring JMeter tests?
- What are the benefits of integrating InfluxDB and Grafana into the load testing stack?
- Which features are provided by Grafana?
- Demonstration of InfluxDB and Grafana using a practice web application
To view the webinar recording, go to:
https://www.rttsweb.com/jmeter-integration-webinar
State of ICS and IoT Cyber Threat Landscape Report 2024 previewPrayukth K V
The IoT and OT threat landscape report has been prepared by the Threat Research Team at Sectrio using data from Sectrio, cyber threat intelligence farming facilities spread across over 85 cities around the world. In addition, Sectrio also runs AI-based advanced threat and payload engagement facilities that serve as sinks to attract and engage sophisticated threat actors, and newer malware including new variants and latent threats that are at an earlier stage of development.
The latest edition of the OT/ICS and IoT security Threat Landscape Report 2024 also covers:
State of global ICS asset and network exposure
Sectoral targets and attacks as well as the cost of ransom
Global APT activity, AI usage, actor and tactic profiles, and implications
Rise in volumes of AI-powered cyberattacks
Major cyber events in 2024
Malware and malicious payload trends
Cyberattack types and targets
Vulnerability exploit attempts on CVEs
Attacks on counties – USA
Expansion of bot farms – how, where, and why
In-depth analysis of the cyber threat landscape across North America, South America, Europe, APAC, and the Middle East
Why are attacks on smart factories rising?
Cyber risk predictions
Axis of attacks – Europe
Systemic attacks in the Middle East
Download the full report from here:
https://sectrio.com/resources/ot-threat-landscape-reports/sectrio-releases-ot-ics-and-iot-security-threat-landscape-report-2024/
Builder.ai Founder Sachin Dev Duggal's Strategic Approach to Create an Innova...Ramesh Iyer
In today's fast-changing business world, Companies that adapt and embrace new ideas often need help to keep up with the competition. However, fostering a culture of innovation takes much work. It takes vision, leadership and willingness to take risks in the right proportion. Sachin Dev Duggal, co-founder of Builder.ai, has perfected the art of this balance, creating a company culture where creativity and growth are nurtured at each stage.
Software Delivery At the Speed of AI: Inflectra Invests In AI-Powered QualityInflectra
In this insightful webinar, Inflectra explores how artificial intelligence (AI) is transforming software development and testing. Discover how AI-powered tools are revolutionizing every stage of the software development lifecycle (SDLC), from design and prototyping to testing, deployment, and monitoring.
Learn about:
• The Future of Testing: How AI is shifting testing towards verification, analysis, and higher-level skills, while reducing repetitive tasks.
• Test Automation: How AI-powered test case generation, optimization, and self-healing tests are making testing more efficient and effective.
• Visual Testing: Explore the emerging capabilities of AI in visual testing and how it's set to revolutionize UI verification.
• Inflectra's AI Solutions: See demonstrations of Inflectra's cutting-edge AI tools like the ChatGPT plugin and Azure Open AI platform, designed to streamline your testing process.
Whether you're a developer, tester, or QA professional, this webinar will give you valuable insights into how AI is shaping the future of software delivery.
Generating a custom Ruby SDK for your web service or Rails API using Smithyg2nightmarescribd
Have you ever wanted a Ruby client API to communicate with your web service? Smithy is a protocol-agnostic language for defining services and SDKs. Smithy Ruby is an implementation of Smithy that generates a Ruby SDK using a Smithy model. In this talk, we will explore Smithy and Smithy Ruby to learn how to generate custom feature-rich SDKs that can communicate with any web service, such as a Rails JSON API.
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
Securing your Kubernetes cluster_ a step-by-step guide to success !KatiaHIMEUR1
Today, after several years of existence, an extremely active community and an ultra-dynamic ecosystem, Kubernetes has established itself as the de facto standard in container orchestration. Thanks to a wide range of managed services, it has never been so easy to set up a ready-to-use Kubernetes cluster.
However, this ease of use means that the subject of security in Kubernetes is often left for later, or even neglected. This exposes companies to significant risks.
In this talk, I'll show you step-by-step how to secure your Kubernetes cluster for greater peace of mind and reliability.
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
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
PN
QN
P
Q
A rational expression is reduced (simplified) if all common
factors are cancelled.
Cancellation Law: Common factor may be cancelled as 1, i.e.
52. PN
QN
= =
1
Rational Expressions
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.
Cancellation Law: Common factor may be cancelled as 1, i.e.
53. PN
QN
= =
1
Rational Expressions
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
Cancellation Law: Common factor may be cancelled as 1, i.e.
54. PN
QN
= =
1
Rational Expressions
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
Cancellation Law: Common factor may be cancelled as 1, i.e.
55. PN
QN
= =
1
Rational Expressions
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
Cancellation Law: Common factor may be cancelled as 1, i.e.
56. PN
QN
= =
1
Rational Expressions
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.
Cancellation Law: Common factor may be cancelled as 1, i.e.
57. PN
QN
= =
1
Rational Expressions
Cancellation Law: Common factor may be cancelled as 1, 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.