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This presentation provides a drill on addition or subtraction of monomials as a practice on the beginning of the slides. It also presents the definition of sequence, arithmetic and geometric sequence with their examples and an activity to perform.
The presentation has first a drill on signed numbers. Then, it provides a definition examples and activities for the topics, " Finding the nth term of an Arithmetic Sequence, Arithmetic Mean and Arithmetic Series.".
This presentation provides a drill on addition or subtraction of monomials as a practice on the beginning of the slides. It also presents the definition of sequence, arithmetic and geometric sequence with their examples and an activity to perform.
The presentation has first a drill on signed numbers. Then, it provides a definition examples and activities for the topics, " Finding the nth term of an Arithmetic Sequence, Arithmetic Mean and Arithmetic Series.".
Fjfjdjejrkdidjxjsjrkkdjdkkjjkkfkckckdkükememememdjdjdjjdnrnene3irjrjrjüjejsjdndnenejejdjdjdjdjdjsjsjejrururirjrjrjddjdjjdjjrjrjrjrjriejrjrjridududjrjrjejejeididjdjdje**Introduction:**
If you want to focus better and get more things done, there are two simple solutions to consider. First, turn off notifications on your phone, especially from news apps and emails. Second, try deleting Facebook from your smartphone – it can help you avoid wasting time.
**Body:**
1. *Turn Off Notifications:*
To concentrate on your tasks, start by stopping notifications from bothering you. Turn off those pop-up messages from news apps and emails. This way, you won't be interrupted while working or studying.
2. *Delete Facebook:*
Consider removing Facebook from your phone. It might seem difficult at first, but it can help you stop spending too much time scrolling through posts. This small change can give you more free time.
**Conclusion:**
These ideas won't magically make you super productive, but they can make your life a bit simpler. By managing notifications and removing distractions like Facebook, you can focus on what's important in your life.**Introduction:**
If you want to focus better and get more things done, there are two simple solutions to consider. First, turn off notifications on your phone, especially from news apps and emails. Second, try deleting Facebook from your smartphone – it can help you avoid wasting time.
**Body:**
1. *Turn Off Notifications:*
To concentrate on your tasks, start by stopping notifications from bothering you. Turn off those pop-up messages from news apps and emails. This way, you won't be interrupted while working or studying.
2. *Delete Facebook:*
Consider removing Facebook from your phone. It might seem difficult at first, but it can help you stop spending too much time scrolling through posts. This small change can give you more free time.
**Conclusion:**
These ideas won't magically make you super productive, but they can make your life a bit simpler. By managing notifications and removing distractions like Facebook, you can focus on what's important in your life.**Introduction:**
If you want to focus better and get more things done, there are two simple solutions to consider. First, turn off notifications on your phone, especially from news apps and emails. Second, try deleting Facebook from your smartphone – it can help you avoid wasting time.
**Body:**
1. *Turn Off Notifications:*
To concentrate on your tasks, start by stopping notifications from bothering you. Turn off those pop-up messages from news apps and emails. This way, you won't be interrupted while working or studying.
2. *Delete Facebook:*
Consider removing Facebook from your phone. It might seem difficult at first, but it can help you stop spending too much time scrolling through posts. This small change can give you more free time.
**Conclusion:**
These ideas won't magically make you super productive, but they can make your life a bit
You will learn how to get the value of a, b and c given a quadratic equations.
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Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
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
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
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3. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
Exponentes y
Productos
Notables
En esta semana repasamos los
temas de leyes de exponentes y
productos notables, necesarios
para los siguientes temas del
curso.
𝑎𝑚
. 𝑎𝑛
= 𝑎𝑚+𝑛
𝑎 + 𝑏 2
= 𝑎2
+ 2𝑎𝑏 + 𝑏2
4. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
LEYES DE EXPONENTES
EXPONENTE NATURAL 𝑎𝑛
= 𝑎. 𝑎. 𝑎 … 𝑎
𝑛 𝑣𝑒𝑐𝑒𝑠
; 𝑛 ∈ ℕ 54
= 5.5.5.5 = 625
EXPONENTE CERO 𝑎0 = 1; 𝑎 ≠ 0 −
2
3
0
= 1
EXPONENTE NEGATIVO 𝑎−𝑛 =
1
𝑎𝑛
2
5
−3
=
5
2
3
=
53
23 =
125
8
EXPONENTE FRACCIONARIO 𝑎
𝑚
𝑛 =
𝑛
𝑎𝑚 8
2
3 =
3
82 =
3
8
2
= 22
5. C R E E M O S E N L A E X I G E N C I A
TEOREMAS
C U R S O D E Á L G E B R A
𝒂𝒎
. 𝒂𝒏
= 𝒂𝒎+𝒏
𝒂𝒎 𝒏
= 𝒂𝒎.𝒏
𝒂. 𝒃 𝒏
= 𝒂𝒏
. 𝒃𝒏
𝒂
𝒃
𝒏
=
𝒂𝒏
𝒃𝒏
𝒂𝒎
𝒂𝒏 = 𝒂𝒎−𝒏
𝒎 𝒏
𝒂 = 𝒎.𝒏
𝒂
𝒏
𝒂. 𝒃 = 𝒏
𝒂.
𝒏
𝒃
𝒏 𝒂
𝒃
=
𝒏
𝒂
𝒏
𝒃
𝑎7. 𝑎5 = 𝑎7+5 = 𝑎12 𝑎8
𝑎3
= 𝑎8−3 = 𝑎5
𝑎7 4 = 𝑎7.4 = 𝑎28
2𝑥 5
= 25
. 𝑥5
2
3
5
=
25
35
=
32
243
3 4
𝑎 = 3.4
𝑎 = 12
𝑎
3
8𝑥 =
3
8. 3
𝑥 = 23
𝑥
3 8
125
=
3
8
3
125
=
2
5
Ejemplos: Ejemplos:
TEOREMAS
6. C R E E M O S E N L A E X I G E N C I A
TEOREMAS
C U R S O D E Á L G E B R A
5
𝑥3 4
𝑥5
𝒂
𝒙𝒎𝒃
𝒙𝒏 =
𝒂𝒃
𝒙𝒎𝒃+𝒏
Ejemplo:
RADICALES INFINITOS
𝑴 =
𝒏
𝒂
𝒏
𝒂 𝒏
𝒂 …
𝒊𝒏𝒇𝒊𝒏𝒊𝒕𝒐𝒔 𝒓𝒂𝒅𝒊𝒄𝒂𝒍𝒆𝒔
Ejemplo: Calcule el valor de T
𝑇 =
3
25
3
25
3
25 …
𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑜𝑠 𝑟𝑎𝑑𝑖𝑐𝑎𝑙𝑒𝑠
=
3
25. 𝑇
𝑇3
𝑇
= 25 𝑇 = 5
=
5.4
𝑥3.4+5 =
20
𝑥17
→ 𝑴 =
𝒏
𝒂. 𝑴 → 𝑇3
= 25𝑇
→ 𝑇2
= 25
7. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
RADICALES INFINITOS
𝑷 =
𝒏
𝒂 +
𝒏
𝒂 + 𝒏
𝒂 + ⋯
𝒊𝒏𝒇𝒊𝒏𝒊𝒕𝒐𝒔 𝒓𝒂𝒅𝒊𝒄𝒂𝒍𝒆𝒔
Ejemplo:
Calcule el valor de J
𝐽 = 12 + 12 + 12 + ⋯
𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑜𝑠 𝑟𝑎𝑑𝑖𝑐𝑎𝑙𝑒𝑠
Solución:
𝐽 = 12 + 12 + 12 + ⋯
𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑜𝑠 𝑟𝑎𝑑𝑖𝑐𝑎𝑙𝑒𝑠
= 12 + 𝐽
𝐽2
= 12 + 𝐽
𝐽 − 4 𝐽 + 3 = 0
Como 𝐽 > 0
→ 𝑷 =
𝒏
𝒂 + 𝑷
→ 𝐽 = 12 + 𝐽
→ 𝐽2
− 𝐽 − 12 = 0
→ 𝐽 = 4 ∨ 𝐽 = −3
→ 𝐽 = 4
8. C R E E M O S E N L A E X I G E N C I A
PRODUCTOS NOTABLES
C U R S O D E Á L G E B R A
PRODUCTO DE BINOMIOS CON TÉRMINO COMÚN
𝒙 + 𝒂 𝒙 + 𝒃 = 𝑥 + 5 𝑥 + 8 =
BINOMIO AL CUADRADO
𝒂 + 𝒃 𝟐 =
𝒂 − 𝒃 𝟐
=
3𝑥 + 5𝑦 2
=
= 9𝑥2
+ 30𝑥𝑦 + 25𝑦2
6 − 2
2
=
= 6 − 4 6 + 4
Ejemplos:
+ 𝒂 + 𝒃 𝒙
𝒙𝟐
+𝒂𝒃 𝑥2
+ 5 + 8 𝑥 +5.8 = 𝑥2 + 13𝑥 + 40
𝒂𝟐
+ 𝟐𝒂𝒃 + 𝒃𝟐
𝒂𝟐
− 𝟐𝒂𝒃 + 𝒃𝟐
3𝑥 2 + 2 5𝑦 2
3𝑥 5𝑦 +
6
2
−2 6 2 + 2 2
= 10 − 4 6
9. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
IDENTIDADES DE LEGENDRE
𝒂 + 𝒃 𝟐
+ 𝒂 − 𝒃 𝟐
=
𝒂 + 𝒃 𝟐 − 𝒂 − 𝒃 𝟐 =
6 + 5
2
+ 6 − 5
2
= 2 6 + 25
7 + 3
2
− 7 − 3
2
=
DIFERENCIA DE CUADRADOS
𝒂 + 𝒃 𝒂 − 𝒃 = 3𝑥 + 2𝑦 3𝑥 − 2𝑦
Ejemplos:
𝟐 𝒂𝟐
+ 𝒃𝟐
𝟒𝒂𝒃
= 2 6
2
+ 52
= 2 31 = 62
4 7 3 = 4 21
𝒂𝟐 − 𝒃𝟐 = 3𝑥 2
= 9𝑥2 − 4𝑦2
− 2𝑦 2
10. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
TRINOMIO AL CUADRADO
𝒂 + 𝒃 + 𝒄 𝟐
𝑎𝑏 + 𝑏𝑐 + 𝑎𝑐 2
BINOMIO AL CUBO
𝒂 + 𝒃 𝟑 𝑥 + 2 3
𝒂 − 𝒃 𝟑
𝑥 − 1 3
IDENTIDADES DE CAUCHY
𝒂 + 𝒃 𝟑 =
𝒂 − 𝒃 𝟑 =
𝑥 + 2 3
=
𝑥 − 1 3 =
Ejemplos:
Ejemplos:
Ejemplos:
= 𝒂𝟐
+ 𝒃𝟐
+ 𝒄𝟐 +𝟐 𝒂𝒃 + 𝒃𝒄 + 𝒄𝒂 = 𝑎𝑏 2
+ 𝑏𝑐 2
+ 𝑎𝑐 2
+2𝑎𝑏𝑐 𝑎 + 𝑏 + 𝑐
= 𝒂𝟑 + 𝟑𝒂𝟐𝒃 + 𝟑𝒂𝒃𝟐 + 𝒃𝟑
= 𝒂𝟑 − 𝟑𝒂𝟐𝒃 + 𝟑𝒂𝒃𝟐 − 𝒃𝟑
= 𝑥3
+ 3. 𝑥2
. 2 + 3. 𝑥. 22
+ 23
= 𝑥3 + 6𝑥2 + 12𝑥 + 8
= 𝑥3
− 3. 𝑥2
. 1 + 3. 𝑥. 12
− 13
= 𝑥3
− 3𝑥2
+ 3𝑥 − 1
𝒂𝟑
+ 𝒃𝟑
+ 𝟑𝒂𝒃 𝒂 + 𝒃
𝒂𝟑
− 𝒃𝟑
− 𝟑𝒂𝒃 𝒂 − 𝒃
𝑥3
+ 23
+ 3. 𝑥. 2. (𝑥 + 2)
𝑥3 − 13 − 3. 𝑥. 1. 𝑥 − 1
11. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
SUMA Y DIFERENCIA DE CUBOS
𝑎3
+ 𝑏3
=
𝑎3 − 𝑏3 =
𝑥 − 1 𝑥2 + 𝑥 + 1 =
𝑥 + 1 𝑥2
− 𝑥 + 1 =
IGUALDADES CONDICIONALES
Si: 𝑎 + 𝑏 + 𝑐 = 0
𝑎2 + 𝑏2 + 𝑐2 = −2 𝑎𝑏 + 𝑏𝑐 + 𝑎𝑐
𝑎3 + 𝑏3 + 𝑐3 = 3𝑎𝑏𝑐
Ejemplos:
Si 𝑎 = 4 − 7; 𝑏 = 7 − 1; 𝑐 = −3
Calcule el valor de
𝑀 =
𝑎3 + 𝑏3 + 𝑐3
𝑎𝑏
Resolución:
𝑎 + 𝑏 + 𝑐 = 0
Entonces, lo que nos piden queda:
𝑎3
+ 𝑏3
+ 𝑐3
𝑎𝑏
𝑎 + 𝑏
𝑎 − 𝑏
𝑎2
− 𝑎𝑏 + 𝑏2
𝑎2 + 𝑎𝑏 + 𝑏2
𝑥3
− 1
𝑥3 + 1
De los datos, se observa que
→ 𝑎3 + 𝑏3 + 𝑐3 = 3𝑎𝑏𝑐
=
3𝑎𝑏𝑐
𝑎𝑏
= 3𝑐 = 3 −3 = −9
12. C R E E M O S E N L A E X I G E N C I A
IGUALDADES CONDICIONALES
C U R S O D E Á L G E B R A
Si a, b, c son números reales, tales que:
𝒂𝟐
+ 𝒃𝟐
+ 𝒄𝟐
= 𝟎
Ejemplo
Si 𝑎, 𝑏 , 𝑐 son números reales y además
𝑎2
+ 𝑏2
+ 𝑐2
= 2𝑎 + 4𝑏 + 6𝑐 − 14
Calcule el valor de
𝑎 + 𝑏 + 𝑐.
Resolución
Todo al primer miembro, tenemos:
𝑎2 + 𝑏2 + 𝑐2 − 2𝑎 − 4𝑏 − 6𝑐 + 14 = 0
𝑎2
− 2𝑎 + 𝑏2
− 4𝑏 + 𝑐2
− 6𝑐 + 14 = 0
𝑎2
− 2𝑎
𝑎 − 1 2
𝑎 − 1 = 0 ∧ 𝑏 − 2 = 0 ∧ 𝑐 − 3 = 0
𝑎 = 1 ∧ 𝑏 = 2 ∧ 𝑐 = 3
∴ 𝑎 + 𝑏 + 𝑐 = 1 + 2 + 3 = 6
↔ 𝒂 = 𝒃 = 𝒄 = 𝟎
+12
+𝑏2
− 4𝑏+22 +𝑐2 − 6𝑐 +32 +14 −12
−22 −32 = 0
+ 𝑏 − 2 2
+ 𝑐 − 3 2 = 0
13. C R E E M O S E N L A E X I G E N C I A
IGUALDADES CONDICIONALES
C U R S O D E Á L G E B R A
Si a, b, c son números reales, tales que:
𝒂𝟐
+ 𝒃𝟐
+ 𝒄𝟐
= 𝒂𝒃 + 𝒃𝒄 + 𝒄𝒂
Ejemplo
Si 𝑎, 𝑏, 𝑐 son números reales y además
𝑎2 + 𝑏2 + 4 − 𝑎𝑏 − 2𝑎 − 2𝑏 = 0
Calcule el valor de
𝑎2 + 𝑏2
Resolución
Tenemos:
𝑎2 + 𝑏2 + 4 − 𝑎𝑏 − 2𝑎 − 2𝑏 = 0
𝑎2
+ 𝑏2
+ 4 = 𝑎𝑏 + 2𝑎 + 2𝑏
𝑎2
+ 𝑏2
+ 22
= 𝑎𝑏 + 2𝑎 + 2𝑏
𝑎 = 𝑏 = 2
Luego:
𝑎2 + 𝑏2 = 22 + 22 = 4 + 4 = 8
∴
↔ 𝒂 = 𝒃 = 𝒄
14. w w w . a c a d e m i a c e s a r v a l l e j o . e d u . p e