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If you are looking for math video tutorials (with voice recording), you may download it on our YouTube Channel. Don't forget to SUBSCRIBE for you to get updated on our upcoming videos.
https://tinyurl.com/y9muob6q
Also, please do visit our page, LIKE and FOLLOW us on Facebook!
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Basic algebra, trig and calculus needed for physics.
**More good stuff available at:
www.wsautter.com
and
http://www.youtube.com/results?search_query=wnsautter&aq=f
Basic algebra, trig and calculus needed for physics.
**More good stuff available at:
www.wsautter.com
and
http://www.youtube.com/results?search_query=wnsautter&aq=f
Strategic Intervention Material (SIM) was provided for Grade 10 students to enhance learning and to motivate and stir up the attention and interest of the students until they master the topic. This material depicts the entire definition of learning since it concludes a systematic development of students’ comprehension on a distinct lesson in Mathematics 10.
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
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.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
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
Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
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Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
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.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
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!
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
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
OBJETIVOS
𝑃 𝑥
𝑓(𝑥)
𝑒𝑠 𝑒𝑥𝑎𝑐𝑡𝑎 → 𝑓 𝑥 𝑒𝑠 𝑓𝑎𝑐𝑡𝑜𝑟 𝑑𝑒 𝑃 𝑥
✓ Utilizar los criterios de factorización.
Para resolver grandes problemas, es necesario
dividirlos en pequeñas partes y luego resolverlos
por separado.
✓Reconocer los factores de un polinomio.
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
FACTORIZACIÓN
Factorizar un polinomio es transformarlo en una
multiplicación indicada de factores primos.
Se trabajará en ℤ, por tanto solo se trabajará con
polinomios de coeficientes enteros.
NOTA
𝑥2
− 9
Ejemplos
𝑥2
−
1
4
𝑥2
− 3
Factor algebraico
Un polinomio 𝑓 𝑥 de grado no nulo, es considerado
factor de otro polinomio 𝑃 𝑥 , si la división:
𝑃 𝑥
𝑓 𝑥
𝑒𝑠 𝑒𝑥𝑎𝑐𝑡𝑎
Es decir
𝑃 𝑥 = 𝑓 𝑥 . 𝑞 𝑥
factores
Ejemplo
De 𝑃 𝑥 = 2 𝑥 + 2 2𝑥 − 3 5𝑥 + 7 , tenemos que
entre sus factores están 𝑥 + 2
= 𝑥 + 3 𝑥 − 3
= 𝑥 +
1
2
𝑥 −
1
2
= 𝑥 + 3 𝑥 − 3
; 2𝑥 − 3 ; 5𝑥 + 7
o una combinación entre estos factores.
5. 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
Ejemplo
Halle el valor de n, para que 𝑥 − 2 𝑠𝑒𝑎 𝑓𝑎𝑐𝑡𝑜𝑟 𝑑𝑒
𝑃 𝑥 = 2𝑥3 − 3𝑥 + 𝑛
Resolución
Como 𝑥 − 2 es factor de 𝑃 𝑥 = 2𝑥3
− 3𝑥 + 𝑛.
Entonces
𝑃 𝑥
𝑥 − 2
es exacta 𝑅 𝑥 = 0
Utilizando el teorema del resto
𝑅 𝑥 = 𝑃 2
0 = 2 2 3
− 3 2 + 𝑛
𝑛 = - 10
Polinomio irreductible
Un polinomio es irreductible, si no puede ser expresado
como la multiplicación de dos o más factores.
Ejemplo
𝑎) 𝐷𝑎𝑑𝑜 𝑃 𝑥 = 𝑥2
− 25, ¿ es irreductible?
𝑃 𝑥 = 𝑥2
− 25
factores
𝑃 𝑥 = 𝑥2
− 25 no es irreductible
NOTA
Todo polinomio de primer grado es irreductible
= 𝑥2 − 52 = 𝑥 + 5 𝑥 − 5
6. 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
Factor primo
Decimos que 𝑓 𝑥 es un factor primo del polinomio 𝑃 𝑥 ,
si verifica:
𝐼) 𝑓 𝑥 es un factor algebraico del polinomio 𝑃 𝑥
𝐼𝐼) 𝑓 𝑥 es un polinomio irreductible
Ejemplo
Si 𝑃 𝑥 = 3𝑥 − 2 5
5𝑥 − 1 2
7𝑥 + 9 , tenemos que
sus factores primos son:
3𝑥 − 2
Ejemplo
𝑃 𝑥; 𝑦; 𝑧 = 7𝑥2𝑦𝑧3 𝑥𝑦 + 1 𝑥 + 𝑦 + 𝑧
Dado el polinomio
¿Cuántos factores primos tiene y cuáles son?
Resolución
Sus factores primos son
𝑥; 𝑦; 𝑧; 𝑥𝑦 + 1 ; 𝑥 + 𝑦 + 𝑧
En total tiene 5 factores primos
NOTA
𝑥2
El factor no es primo, puesto que 𝑥2 = 𝑥. 𝑥
; 5𝑥 − 1 ; 7𝑥 + 9
8. 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
Factor común/ agrupación
I) Busca un término común.
𝑃 𝑥; 𝑦 = 𝑥𝑦 + 2𝑦
Resolución
Ejemplo
Factorice
𝑃 𝑥; 𝑦 = 𝑥 𝑦 + 2 𝑦
𝑃 𝑥; 𝑦 = 𝑦 𝑥 + 2
II) Término común con menor exponente.
Ejemplo
Factorice
𝑃 𝑥; 𝑦 = 𝑥4
𝑦5
+ 3𝑥3
𝑦6
Resolución
𝑃 𝑥; 𝑦 = 𝑥4
𝑦5
+ 3 𝑥3
𝑦6
Menor
exponente
Menor
exponente
𝑃 𝑥; 𝑦 = 𝑥3𝑦5
III) Se agrupa para buscar factor común.
Ejemplo
𝑥4𝑦5
𝑥3𝑦5
= 𝑥
3𝑥3𝑦6
𝑥3𝑦5
= 3𝑦
Factorice
𝑃 𝑥; 𝑦 = 𝑥𝑦 + 2𝑥 + 3𝑦 + 6
𝑃 𝑥; 𝑦 = 𝑥𝑦 + 2𝑥 + 3𝑦 + 6
Resolución
Agrupando tenemos
𝑃 𝑥; 𝑦
𝑃 𝑥; 𝑦
𝑥 + 3𝑦
+3 𝑦 + 2
= 𝑥 𝑦 + 2
= 𝑦 + 2 𝑥 + 3
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
Por identidades
𝑎2 ± 2𝑎𝑏 + 𝑏2 = 𝑎 ± 𝑏 2
𝑎2
− 𝑏2
= 𝑎 + 𝑏 𝑎 − 𝑏
Ejemplo Factorice
𝑃 𝑥; 𝑦 = 𝑥2
+ 4𝑥 + 4 − 𝑦2
Resolución
𝑃 𝑥; 𝑦 = 𝑥2 + 4𝑥 + 4 − 𝑦2
𝑥 + 2 2
𝑃 𝑥; 𝑦 = 𝑥 + 2 2
−𝑦2
𝑃 𝑥; 𝑦 = 𝑥 + 2 + 𝑦 𝑥 + 2 − 𝑦
𝑎3 + 𝑏3 = 𝑎 + 𝑏 𝑎2 − 𝑎𝑏 + 𝑏2
𝑎3
− 𝑏3
= 𝑎 − 𝑏 𝑎2
+ 𝑎𝑏 + 𝑏2
Ejemplo Factorice
𝑃 𝑥; 𝑦 = 𝑥6
− 𝑦6
Resolución
Ejemplo Factorice
𝑃 𝑥 = 𝑥4
+ 6𝑥2
+ 25
Resolución
𝑃 𝑥 = 𝑥4
+ 25 + 6𝑥2
Se busca un TCP
+10𝑥2 −10𝑥2
𝑃 𝑥 = 𝑥2 + 5 2 −4𝑥2
𝑃 𝑥 = 𝑥2
+ 5 2
− (2𝑥)2
𝑃 𝑥 = 𝑥2
+ 5 + 2𝑥 𝑥2
+ 5 − 2𝑥
𝑃 𝑥 = 𝑥2
+ 2𝑥 + 5 𝑥2
− 2𝑥 + 5
𝑃 𝑥; 𝑦 = 𝑥3 2
− 𝑦3 2
𝑃 𝑥; 𝑦 = 𝑥3
+ 𝑦3
𝑥3
− 𝑦3
Suma de cubos Diferencia de cubos
𝑃 𝑥; 𝑦 =
𝑥 + 𝑦 𝑥2
− 𝑥𝑦 + 𝑦2
𝑥 − 𝑦 𝑥2
+ 𝑥𝑦 + 𝑦2
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
Aspa simple
𝑆𝑒 𝑎𝑝𝑙𝑖𝑐𝑎 𝑎 𝑝𝑜𝑙𝑖𝑛𝑜𝑚𝑖𝑜𝑠 𝑑𝑒 𝑙𝑎 𝑓𝑜𝑟𝑚𝑎:
𝑃 𝑥; 𝑦 = 𝐴𝑥2𝑚 + 𝐵𝑥𝑚𝑦𝑛 + 𝐶𝑦2𝑛
Procedimiento
I) Descomponer los extremos convenientemente
𝑃 𝑥; 𝑦 = 𝐴𝑥2𝑚
+ 𝐵𝑥𝑚
𝑦𝑛
+ 𝐶𝑦2𝑛
𝑎1𝑥𝑚
𝑎2𝑥𝑚
𝑐1𝑦𝑛
𝑐2𝑦𝑛
II) Se comprueba que el término central es igual a
la suma de los productos parciales en forma de
aspa
III) Luego
𝑎2𝑐1𝑥𝑚𝑦𝑛
𝑎1𝑐2𝑥𝑚
𝑦𝑛
+
𝐵 = 𝑎2𝑐1 + 𝑎1𝑐2
𝑃 𝑥; 𝑦 =
Factor
Factor
𝑎1𝑥𝑚
+ 𝑐1𝑦𝑛
𝑎2𝑥𝑚
+ 𝑐2𝑦𝑛
Ejemplo 1
𝑃 𝑥 = 3𝑥2
+ 10𝑥 + 8
3𝑥
𝑥
+4
+2
+4𝑥
+6𝑥
+
Factor
Factor
+10𝑥
𝑃 𝑥 = 3𝑥 + 4 𝑥 + 2
Ejemplo 2
𝑃 𝑥; 𝑦 = 15𝑥4
− 11𝑥2
𝑦2
+ 2𝑦4
5𝑥2
3𝑥2
−2𝑦2
−𝑦2
−6𝑥2
𝑦2
−5𝑥2
𝑦2
+
Factor
Factor
−11𝑥2𝑦2
𝑃 𝑥; 𝑦 = 5𝑥2 − 2𝑦2
3𝑥2
− 𝑦2
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
Aspa doble especial
𝑆𝑒 𝑎𝑝𝑙𝑖𝑐𝑎 𝑎 𝑝𝑜𝑙𝑖𝑛𝑜𝑚𝑖𝑜𝑠 𝑑𝑒 𝑐𝑢𝑎𝑟𝑡𝑜 𝑔𝑟𝑎𝑑𝑜, 𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑜𝑠 𝑦 𝑜𝑟𝑑𝑒𝑛𝑎𝑑𝑜𝑠 𝑒𝑛 𝑓𝑜𝑟𝑚𝑎 𝑑𝑒𝑐𝑟𝑒𝑐𝑖𝑒𝑛𝑡𝑒.
Procedimiento
I) Se descomponen los extremos.
𝑃 𝑥 = 𝐴𝑥4
+ 𝐵𝑥3
+ 𝐶𝑥2
+ 𝐷𝑥 + 𝐸
𝑎1𝑥2
𝑎2𝑥2
𝑒1
𝑒2
𝑎2𝑒1𝑥2
𝑎1𝑒2𝑥2
(+)
𝐹𝑥2
(−)
𝑘1𝑥
𝑘2𝑥
𝐾𝑥2
II) Se realiza el aspa simple con los extremos y se obtiene 𝐹𝑥2
.
IV) Se descompone 𝐾𝑥2, de tal manera que cumple las dos aspas simples en ambos lados.
V) Los factores se toma en forma horizontal.
𝑃 𝑥 =
Factor
Factor
III) Se realiza la diferencia 𝐶𝑥2
− 𝐹𝑥2
= 𝐾𝑥2
.
𝑎1𝑥2
+ 𝑘1𝑥 + 𝑒1 𝑎2𝑥2
+ 𝑘2𝑥 + 𝑒2
12. 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
Ejemplo 1
𝑃 𝑥 = 𝑥4
+ 7𝑥3
+ 14𝑥2
+ 7𝑥 + 1
𝑥2
𝑥2
+1
+1
+𝑥2
+𝑥2
(+)
+2𝑥2
(−)
+3𝑥
+4𝑥
+12𝑥2
Factor
Factor
Factorice
𝑃 𝑥 = 𝑥2 + 3𝑥 + 1 𝑥2
+ 4𝑥 + 1
Ejemplo 2
𝑃 𝑥 = 𝑥4 + 𝑥3 + 2𝑥2 + 5𝑥 − 15
𝑥2
𝑥2
+5
-3
+5𝑥2
−3𝑥2
(+)
+2𝑥2
(−)
+0𝑥
+𝑥
+0𝑥2
Factor
Factor
Factorice
𝑃 𝑥 = 𝑥2
+ 0𝑥 + 5 𝑥2
+ 𝑥 − 3
𝑃 𝑥 = 𝑥2
+ 5 𝑥2
+ 𝑥 − 3
13. C R E E M O S E N L A E X I G E N C I A
Criterio de divisores binómicos
Se utiliza para factorizar los polinomios en una variable y de grado superior a dos, siempre y cuando admita por lo
menos un factor lineal.
Raíz de un polinomio
Si P 𝑥 es un polinomio de grado mayor que cero,
decimos que 𝛼 es raíz del polinomio P 𝑥 , sí y solo
sí P 𝛼 = 0
𝑃 𝑥 = 𝑥3 − 3𝑥 − 2
Ejemplo
𝑃 0 = (0)3
−3 0 − 2 = −2 →
𝑃 1 = (1)3−3 1 − 2 = −4 →
𝑃 2 = (2)3
−3 2 − 2 = 0 →
Posibles raíces racionales (P.R.R)
Para conocer las posibles raíces racionales de un polinomio
P 𝑥 de coeficientes enteros.
P 𝑥 = 𝑎0𝑥𝑛
+ 𝑎1𝑥𝑛−1
+ 𝑎2𝑥𝑛−2
+ ⋯ + 𝑎𝑛−1𝑥 + 𝑎𝑛
Se utilizará el siguiente criterio
P. R. R = ±
𝐷𝑖𝑣𝑖𝑠𝑜𝑟𝑒𝑠 𝑑𝑒 𝑎𝑛
𝐷𝑖𝑣𝑖𝑠𝑜𝑟𝑒𝑠 𝑑𝑒 𝑎0
(𝑎0. 𝑎𝑛 ≠ 0)
Ejemplo
𝑃 𝑥 = 3𝑥4
+ 2𝑥2
+ 4𝑥 − 9 → P. R. R = ±
𝐷𝑖𝑣𝑖𝑠𝑜𝑟𝑒𝑠 𝑑𝑒 9
𝐷𝑖𝑣𝑖𝑠𝑜𝑟𝑒𝑠 𝑑𝑒 3
P. R. R = ±
1; 3; 9
1; 3
;
= ± 1; 3; 9;
1
3
0 𝑛𝑜 𝑒𝑠 𝑟𝑎í𝑧
1 𝑛𝑜 𝑒𝑠 𝑟𝑎í𝑧
2 𝑠𝑖 𝑒𝑠 𝑟𝑎í𝑧
14. C R E E M O S E N L A E X I G E N C I A
NOTA
Las posibles raíces racionales (P.R.R), nos muestran los
valores racionales que posiblemente puedan ser raíces
del polinomio con coeficientes enteros.
𝑃 𝑥 = 2𝑥2 − 3𝑥 − 2 tenemos:
𝑃. 𝑅. 𝑅
Si
𝑃. 𝑅. 𝑅 = 1; −1; 2; −2;
1
2
; −
1
2
evaluando
𝑃 1 = −3 𝑃 −1 = 3 𝑃 2 = 0
𝑃 −2 = 12 𝑃
1
2
= −3 𝑃 −
1
2
= 0
No son raíces Son raíces
De los 6 posibles valores, solo 2 son raíces
Teorema del factor
𝛼 es una raíz del polinomio P 𝑥 si y solo si
𝑥 − 𝛼 𝑒𝑠 𝑓𝑎𝑐𝑡𝑜𝑟 𝑑𝑒 𝑃 𝑥
Ejemplo
𝑃 𝑥 = 𝑥3 + 5𝑥 + 6
Si tenemos:
𝑃. 𝑅. 𝑅 = ±
𝐷𝑖𝑣𝑖𝑠𝑜𝑟𝑒𝑠 𝑑𝑒 6
𝐷𝑖𝑣𝑖𝑠𝑜𝑟𝑒𝑠 𝑑𝑒 1
como 𝑃 −1 = −1 3
+ 5 −1 + 6 = 0
-1 es raíz de 𝑃 𝑥
𝑥 − −1 =
𝑃 𝑥 =
NOTA 𝑞(𝑥) se calcula por división (regla de Ruffini)
Criterio de divisores binómicos
= ±
1; 2; 3; 6
1
= ± 1; 2; 3; 6
𝑥 + 1 es un factor de 𝑃 𝑥
𝑥 + 1 𝑞(𝑥)
= ±
𝐷𝑖𝑣𝑖𝑠𝑜𝑟𝑒𝑠 𝑑𝑒 2
𝐷𝑖𝑣𝑖𝑠𝑜𝑟𝑒𝑠 𝑑𝑒 2
= ±
1; 2
1; 2
= ± 1; 2;
1
2
15. C R E E M O S E N L A E X I G E N C I A
Criterio de divisores binómicos
Procedimiento
Dado el polinomio
P 𝑥 = 𝑎0𝑥𝑛
+ 𝑎1𝑥𝑛−1
+ 𝑎2𝑥𝑛−2
+ ⋯ + 𝑎𝑛−1𝑥 + 𝑎𝑛
con coeficientes enteros, donde 𝑎0. 𝑎𝑛 ≠ 0
I) Se halla sus P.R.R que nos permite encontrar una raíz
del polinomio; por teorema del factor, se podrá conocer
un factor.
II) Se hace una división por Ruffini entre el polinomio y el
factor encontrado, siendo el cociente el otro factor
buscado.
Ejemplo
Factorice 𝑃 𝑥 = 𝑥3 − 7𝑥 + 6
Resolución
I) Tenemos
𝑃. 𝑅. 𝑅 = ±
𝐷𝑖𝑣𝑖𝑠𝑜𝑟𝑒𝑠 𝑑𝑒 6
𝐷𝑖𝑣𝑖𝑠𝑜𝑟𝑒𝑠 𝑑𝑒 1
Como
𝑃 1 = (1)3−7 1 + 6 = 0 1 𝑒𝑠 𝑟𝑎í𝑧 𝑑𝑒 𝑃(𝑥)
𝑥 − 1 𝑒𝑠 𝑢𝑛 𝑓𝑎𝑐𝑡𝑜𝑟 𝑑𝑒 𝑃 𝑥
𝑃 𝑥 = 𝑥 − 1 𝑞 𝑥
= ±
1; 2; 3; 6
1
= ± 1; 2; 3; 6
16. C R E E M O S E N L A E X I G E N C I A
II) Encontramos el otro factor por la regla de Ruffini
𝑃 𝑥 ÷ 𝑥 − 1
Criterio de divisores binómicos
Tenemos:
𝑃 𝑥
𝑥 − 1
=
𝑥3
− 7𝑥 + 6
𝑥 − 1
=
𝑥3
+ 0𝑥2
− 7𝑥 + 6
𝑥 − 1
Por la regla de Ruffini, tenemos:
𝑥 − 1 = 0
𝑥 = 1
1 0 −7 6
1
1
1
1
−6
−6
0
𝑞 𝑥
𝑃 𝑥 = 𝑥 − 1 𝑞 𝑥
Recordemos que
𝑃 𝑥 = 𝑥 − 1 𝑥2
+ 𝑥 − 6
Se puede factorizar
por aspa simple
𝑃 𝑥 = 𝑥 − 1 𝑥2
+ 𝑥 − 6
𝑥
𝑥
+3
−2
𝑃 𝑥 = 𝑥 − 1 𝑥 + 3 𝑥 − 2
17. 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