This document provides an introduction to complex numbers. It discusses:
1) The definition of the imaginary number i and its powers.
2) How complex numbers combine real and imaginary numbers in the form of a + bi.
3) The four basic operations that can be performed on complex numbers: addition, subtraction, multiplication, and division.
This Our presentation about complex number.
This is very easy and you can explore it to all within a short time .
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This Our presentation about complex number.
This is very easy and you can explore it to all within a short time .
I think everybody like it and satisfied about this presentation.
The presentation is about Complex numbers: How they originated, what they are and how to do the operations of addition, subtraction, multiplication, and division.
By Dr. Farhana Shaheen
The presentation is about Complex numbers: How they originated, what they are and how to do the operations of addition, subtraction, multiplication, and division.
By Dr. Farhana Shaheen
This tutorial provides fundamental concepts such as:
- Absolute Values
- Basic Operations with Signed Numbers
- PEMDAS rule
in order to properly handle simplification of mathematical expressions.
Mth 163 Test-Unit II Name___________________________
Sections 4.2-4.5, 2.5
1. Identify the left-hand and right-hand behavior of the graph of the polynomial
function f(x) = 6x
3
– 5x + 4.
2. Find a polynomial function that has zeros at -3, -1, and -2. Do not leave it in
factored form, but multiply out your answer.
3. Use the Intermediate Value Theorem to determine an interval bounded by integers
one unit in length in which the polynomial function is guaranteed to have a zero.
[EXPLAIN]
f(x) = 3x
5
– 9x
4
+ 5x
3
– 4x
2
+ x – 3
4. Use synthetic division to determine which of the following is a solution of the
equation 3x
4
– 2x
3
+ 26x
2
– 18x – 9 = 0.
[A] 3 [B] 1 [C] - 3 [D] 1/3 [E] None of these
5. Use synthetic division to divide 4x
4
– 4x
3
-8x – 14 by x – 2. Show your work and
state the quotient and remainder.
6. Factor 2x
3
+ 15x
2
+ 27x +10 given that x+2 is one of the factors. SHOW YOUR
WORK.
7. Determine the maximum number of zeros of the polynomial function
f(x) = 7x
2
+ 7x – 7x +2.
8. Use the Rational Zero Theorem to determine all possible rational zeros of
f(x) = 3x
5
– 6x
3
– 2x
2
+ 9
Do not find the actual zeros.
9. Find all the real zeros of the function f(x) = x
3
+ 2x
2
– 13x + 10. SHOW YOUR
WORK.
10. A polynomial function of degree 5 whose coefficients are real numbers has the
zeros 2, -9i, and -9+i. Identify the remaining zeros.
11. Write this polynomial in completely factored form: 10x
3
+29x
2
+ 4x – 15. SHOW
YOUR WORK.
12. Use the Remainder Theorem, not direct substitution, to find f(-2) when
f(x) = 4x
3
+ 3x + 10. Show your work and specify what f(-2) is.
13. Wouldn’t it be nice to win a million dollars? Most people think so, but one thing
many forget to consider is the amount of taxes that must be paid on the winnings. The
table below shows the relationship between the winnings and the approximate amount
of taxes to be paid to the IRS. Draw a scatter plot to model the data. Then find an
equation for the best-fit line and use the equation to estimate the amount of taxes to
be paid on winnings of $5,300,000.
Amount Won Taxes Due
$470,000 $100,000
$670,000 $180,000
$1,420,000 $480,000
$1,995,000 $710,000
2,545,000 $930,000
...
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.
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?
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.
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.
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.
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.
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
2. 2
The Imaginary Number i
By definition
Consider powers if i
2
1 1i i− = ⇔ = −
2
3 2
4 2 2
5 4
1
1 1 1
1
...
i
i i i i
i i i
i i i i i
= −
= × = −
= × = − ×− =
= × = × =
It's any
number
you can
imagine
3. 3
Using i
Now we can handle quantities that
occasionally show up in mathematical
solutions
What about
1a a i a− = − × =
49− 18−
9. 9
Operations on Complex Numbers
Division technique
Multiply numerator and denominator by the
conjugate of the denominator
3
5 2
i
i−
2
2
3 5 2
5 2 5 2
15 6
25 4
6 15 6 15
29 29 29
i i
i i
i i
i
i
i
+
= ×
− +
+
=
−
− +
= = − +
10. 10
Complex Numbers on the Calculator
Possible result
Reset mode
Complex format
to Rectangular
Now calculator does
desired result
11. 11
Complex Numbers on the Calculator
Operations with complex on calculator
Make sure to use the
correct character for i.
Use 2nd
-i
Make sure to use the
correct character for i.
Use 2nd
-i
12. 12
Warning
Consider
It is tempting to combine them
The multiplicative property of radicals only
works for positive values under the radical sign
Instead use imaginary numbers
16 49− × −
16 49 16 49 4 7 28− ×− = + × = × =
2
16 49 4 7 4 7 28i i i− ×− = × = × × = −
13. 13
Try It Out
Use the correct principles to simplify
the following:
3 121− × −
( ) ( )4 81 4 81+ − × − −
( )
2
3 144− −
