This document contains math exercises involving dividing polynomials and finding quotients and remainders. It includes solving divisions and writing the quotient and remainder, calculating an unknown value m based on a given remainder, applying Ruffini's rule to divide polynomials, and using Ruffini's rule to find m with a given remainder. The exercises cover dividing polynomials, writing quotients and remainders, and using long division to solve for unknowns.
In this short paper, we have provided a new method for finding the square for any positive
integer. Furthermore, a new type of numbers are called T-semi prime numbers are studied by using the prime
numbers.
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This is a Question Papers of Mumbai University for B.Sc.IT Student of Semester - III [Logic and Discrete Mathematics] (Revised Course). [Year - January / 2014] . . .Solution Set of this Paper is Coming soon...
In this short paper, we have provided a new method for finding the square for any positive
integer. Furthermore, a new type of numbers are called T-semi prime numbers are studied by using the prime
numbers.
[Question Paper] Logic and Discrete Mathematics (Revised Course) [January / 2...Mumbai B.Sc.IT Study
This is a Question Papers of Mumbai University for B.Sc.IT Student of Semester - III [Logic and Discrete Mathematics] (Revised Course). [Year - January / 2014] . . .Solution Set of this Paper is Coming soon...
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الاحيائي التطبيقي
باللغة الانكليزية
لمدارس المتميزين والمدارس الأهلية
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B.Sc (Pass) Nautical & Engineering Model Question 2 Mathematics Second Paper
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Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
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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.
4 ESO Academics - Unit 03 - Exercises 4.3.3 - Division of Polynomials. Ruffini´s Rule.
1. UNIT 03 December
1. Solve these divisions and write the remainder and the quotient. Check the
result.
a) (𝟓𝟓𝒙𝒙𝟐𝟐
+ 𝟕𝟕𝟕𝟕 − 𝟑𝟑) ÷ (𝒙𝒙 + 𝟑𝟑)
b) (𝒙𝒙𝟑𝟑
− 𝟐𝟐𝒙𝒙𝟐𝟐
+ 𝟐𝟐𝟐𝟐 − 𝟏𝟏) ÷ (𝒙𝒙 − 𝟏𝟏)
c) (𝟐𝟐𝒙𝒙𝟒𝟒
− 𝟑𝟑𝒙𝒙𝟑𝟑
+ 𝟓𝟓𝟓𝟓 + 𝟒𝟒) ÷ (𝒙𝒙𝟐𝟐
+ 𝟐𝟐)
d) (−𝟑𝟑𝒙𝒙𝟒𝟒
+ 𝟐𝟐𝒙𝒙𝟑𝟑
+ 𝟓𝟓𝟓𝟓 + 𝟒𝟒) ÷ (𝒙𝒙𝟐𝟐
− 𝟑𝟑)
Axel Cotón Gutiérrez Mathematics 4º ESO Exercises – 4.3.3.1
2. UNIT 03 December
2. Calculate m knowing that the remainder of the division is 5.
(𝟑𝟑𝒙𝒙𝟒𝟒
− 𝒎𝒎) ÷ (𝒙𝒙𝟐𝟐
+ 𝟏𝟏)
3. Apply Ruffini´s rule to solve these divisions:
a) (𝟐𝟐𝒙𝒙𝟑𝟑
− 𝟒𝟒𝒙𝒙𝟐𝟐
− 𝟔𝟔𝟔𝟔 + 𝟖𝟖) ÷ (𝒙𝒙 + 𝟏𝟏)
b) (−𝟓𝟓𝒙𝒙𝟑𝟑
+ 𝟐𝟐𝒙𝒙𝟐𝟐
− 𝒙𝒙 + 𝟑𝟑) ÷ (𝒙𝒙 − 𝟐𝟐)
c) (𝒙𝒙𝟒𝟒
+ 𝟑𝟑𝒙𝒙𝟑𝟑
− 𝟓𝟓𝒙𝒙𝟐𝟐
+ 𝒙𝒙 − 𝟕𝟕) ÷ (𝒙𝒙 + 𝟑𝟑)
d) (𝟐𝟐𝒙𝒙𝟒𝟒
− 𝒙𝒙𝟑𝟑
+ 𝟔𝟔𝒙𝒙𝟐𝟐
− 𝟑𝟑𝟑𝟑 + 𝟏𝟏) ÷ (𝒙𝒙 + 𝟐𝟐)
Axel Cotón Gutiérrez Mathematics 4º ESO Exercises – 4.3.3.2
3. UNIT 03 December
e) (𝒙𝒙𝟒𝟒
− 𝒙𝒙𝟑𝟑
+ 𝒙𝒙𝟐𝟐
− 𝒙𝒙 + 𝟏𝟏) ÷ (𝒙𝒙 − 𝟏𝟏)
f) �−𝒙𝒙𝟓𝟓
+ 𝟐𝟐𝒙𝒙𝟒𝟒
− 𝒙𝒙𝟑𝟑
+ 𝟐𝟐𝒙𝒙𝟐𝟐
− 𝒙𝒙 − 𝟏𝟏� ÷ (𝒙𝒙 − 𝟑𝟑)
4. Indicate the quotient and the remainder of these divisions:
a) (𝟓𝟓𝒙𝒙𝟒𝟒
+ 𝟑𝟑𝟑𝟑 − 𝟓𝟓) ÷ (𝒙𝒙 + 𝟏𝟏)
b) (𝟐𝟐𝒙𝒙𝟐𝟐
+ 𝒙𝒙 + 𝟒𝟒 − 𝟑𝟑𝒙𝒙𝟑𝟑) ÷ (𝒙𝒙 − 𝟓𝟓)
c) �−𝒙𝒙𝟑𝟑
+ 𝟐𝟐𝒙𝒙𝟓𝟓
+ 𝟏𝟏� ÷ (𝒙𝒙 − 𝟏𝟏)
Axel Cotón Gutiérrez Mathematics 4º ESO Exercises – 4.3.3.3
4. UNIT 03 December
d) (𝟓𝟓𝟓𝟓 − 𝟑𝟑𝒙𝒙𝟐𝟐
+ 𝟐𝟐𝒙𝒙𝟑𝟑) ÷ (𝒙𝒙 + 𝟐𝟐)
5. Using Ruffini´s rule, find m to have −𝟒𝟒 as remainder of the division:
(𝟕𝟕𝒙𝒙𝟑𝟑
− 𝟓𝟓𝟓𝟓𝟓𝟓 − 𝟐𝟐) ÷ (𝒙𝒙 + 𝟏𝟏)
Axel Cotón Gutiérrez Mathematics 4º ESO Exercises – 4.3.3.4
