This document discusses binomial expansion using Pascal's triangle and the binomial theorem. It explains that a binomial is a polynomial with two terms and the binomial theorem allows for quick expansion of binomial expressions raised to a power. It provides examples of using the binomial theorem to expand binomial expressions up to the x^3 term for different types of binomial questions, including a standard binomial, fractional binomial, and binomial with a square root.
Getting Social Media to Work, So Social Media Works for You - IGSHPA 2017MarketSnare
Created and presented by Kevin R. Mullett (@kmullett) at the 2017 International Ground Source Heat Pump Association Conference in Denver, CO on Wednesday, March 14th.
@igshpa #IGSHPAconf
Discussing how to get social media marketing to work for small businesses.
Control interno y gobierno corporativo riesgo de fraude parte 2Control Interno
Para dar un tono de responsabilidad compartida, la junta directiva en principio debería asegurarse de que los miembros de la Junta estén apropiadamente gobernados.
La Junta tiene debe asegurar que la gerencia diseñe documentación efectiva de la gestión del riesgo de fraude, para incentivar el comportamiento ético y compromiso de los empleados, clientes y proveedores, para que se empeñen en que los estándares se cumplan todos los días.
Based on the readings and content for this course.docxBased on.docxikirkton
Based on the readings and content for this course.docx
Based on the readings and content for this course, which topic did you find most useful or interesting? How will you use it later in life? What makes it valuable?
Ans:
Well, there are many small but significant things in real life, that I understand, are related to maths.
For example the screen size of a TV or a laptop. When we say 14 inch laptop we mean the diagonal of the screen is approximately 14 inches.
This is a direct application of Pythagorean Theorem.
And depending on the shape of the room determines the formula for area that I use. For example, if our room was a perfect square (which none of them are) I would utilize the formula a = s^2, since our rooms our rectangle, the formula we more commonly use is a = lw.
Again I understand the amount of money we spend on gas can be modeled by a mathematical function.
When we throw a ball up, the time taken for it to come down can be modeled by a quadratic equation.
There are so many things.
I won't pick up any particular thing.
But after this course, I am able to look at many things in a more analytical way.
I can understand the mathematical logic behind them.
So all these are valuable to me
Do you always use the property of distribution when multiplying monomials and polynomials.docx
Do you always use the property of distribution when multiplying monomials and polynomials? Explain why or why not. In what situations would distribution become important? Provide an example for the class to practice with.
Ans:
The property of distribution is one important tool in solving the polynomial multiplications.
For example:
3x*(x+5) = 3x*x + 3x*5 = 3x^2 +15x
But this property is used only when one of the bracketed terms contains two or more terms of different order.
For example in the above case, the bracket consists of two different order terms, x and 5.
Let’s take another case:
3x*(x+2x)
This can be solved in two ways.
3x(x+2x) = 3x*x + 3x*2x = 3x^2 +6x^2 = 9x^2
Or we can say:
3x(x+2x) = 3x*3x = 9x^2
So in the 1st method we used the distribute property.
But in the 2nd case we did not.
So it depends on the particular problem and looking at the different terms we can decide whether or not to use distributive property.
Explain how to factor the following trinomials forms.docx
Explain how to factor the following trinomials forms: x2 + bx + c and ax2 + bx + c. Is there more than one way to factor this? Show your answer using both words and mathematical notation
Ans:
Actually I am not very sure how to answer this.
For me x2 + bx + c and ax2 + bx + c are not different from each other.
The 1st one is a special case of the second expression where a=1.
To factor the expression ax^2 + bx + c we first need to factor the middle term bx cleverly.
Now it's to be understood that not all trinomials can be factored. But some of them can be.
Basically, we have to write b in the form b= p+q so that p*q= a*c. This is the only trick.
Then ...
This is a PPT created and developed by Alankrit Wadhwa of Army Public School, Pune. He made this PPT with great effort and is credible for the same. I hope this PPT makes this chapter a lot more fun and easier to understand.
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.
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.
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An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
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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.
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Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
A Strategic Approach: GenAI in EducationPeter 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.
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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.
1. Binomial expansion
Summary:
What is a binomial?
A binomial is a polynomial with two terms (polynomial just means many terms).
This is a binomial
Pascal's Triangle
Pascal's Triangle is 1 way of expanding a binomial.
To build the triangle, start with "1" at the top, then continue placing numbers
below it in a triangular pattern.
The two numbers adding with each other should be equal to the number below
Binomial theorem
The Binomial Theorem is a quick way of expanding a binomial expression that has
been raised to some power. This power is usually something inconveniently large.
Most people prefer to use this method of expanding than Pascal's Triangle since
it’s much quicker. In addition, pascal’s triangle is used for small powers.
2. Type of Binomial Questions
for part A, you should use the binomial theorem on (1+2x) ^3 (Equation stated
above). You can use Pascale’s triangle to work out this equation but it would be
time consuming.
Replace n with 3 and x with 2x
Since the question has asked us to expand up to x^3, any values beyond this
will not be included in our answer
2) for part B, Our binomial is a fraction. For these type of questions, you should
inverse our binomial. This will give us (1-x) ^-1
After you have inversed the binomial, you can now use the binomial theorem to
find our terms up to x^3
Replace n with -1 and x with –x
Any values beyond x^3 will not be included in our answer
3. 3) For part C, our binomial has a square root. From our knowledge of roots, we can
write this as (1+x) ^1/2
Now that our binomial is in index form, we can use binomial theorem to work out
terms up to x^3
Replace n with ½ and x with x
