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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.".
Suma, Resta y Valor numérico de Expresiones algebraicas.
Multiplicación y División de Expresiones algebraicas.
Productos Notables de Expresiones algebraicas.
Factorización por Productos Notables.
Newton Cotes Integration Method, Open Newton Cotes, Closed Newton Cotes Gauss...HadiaZahid2
the description about weddle's rule and newton cotes method and Gaussian quadrature method of numerical computing course.It contains introduction , Rules and Examples
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.".
Suma, Resta y Valor numérico de Expresiones algebraicas.
Multiplicación y División de Expresiones algebraicas.
Productos Notables de Expresiones algebraicas.
Factorización por Productos Notables.
Newton Cotes Integration Method, Open Newton Cotes, Closed Newton Cotes Gauss...HadiaZahid2
the description about weddle's rule and newton cotes method and Gaussian quadrature method of numerical computing course.It contains introduction , Rules and Examples
En este archivo se muestran las consideraciones preliminares para entender limites, tal como factorización, racionalización y valor absoluto. El tema es iniciado con la definición intuitiva, los diferentes teoremas que se aplican en límites, la indeterminación 0/0 y los diversos ejemplos al respecto
SEQUENCE AND SERIES
SEQUENCE
Is a set of numbers written in a definite order such that there is a rule by which the terms are obtained. Or
Is a set of number with a simple pattern.
Example
1. A set of even numbers
• 2, 4, 6, 8, 10 ……
2. A set of odd numbers
• 1, 3, 5, 7, 9, 11….
Knowing the pattern the next number from the previous can be obtained.
Example
1. Find the next term from the sequence
• 2, 7, 12, 17, 22, 27, 32
The next term is 37.
2. Given the sequence
• 2, 4, 6, 8, 10, 12………
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Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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!
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
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
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
POLINOMIOS
Es cotidiano conocer un resultado
para tomar una buena decisión.
Una de las formas más utilizadas
es el uso de los polinomios para
generar modelos matemáticos.
𝒅 = 𝒗𝟎𝒕 +
𝟏
𝟐
𝒂𝒕𝟐
4. C R E E M O S E N L A E X I G E N C I A
Variable: Valor que cambia, no es fijo
Constante: Valor fijo, no cambia
Notación matemática: Es la representación
simbólica que nos permite reconocer cuales
son las variables de una expresión
matemática.
x, y, z
2; 1/3; π
P(x); Q(x; y)
C U R S O D E Á L G E B R A
Ejemplos:
𝑃 𝑥; 𝑦
𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠
= 2𝑥2 + 3𝑥𝑦 − 5𝑦2
𝑀 2𝑥 − 1
𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒
= 5𝑥 + 4
Un polinomio es una expresión que enlaza variables y
constantes mediante una combinación finita de
adiciones, sustracciones, multiplicación y
potenciaciones, en las cuales los exponentes de las
variables son enteros no negativos.
POLINOMIO VARIABLES N° DE TÉRMINOS
𝑅 𝑥 = −7𝑥5 𝑥 1
𝑃 𝑥 = 𝑥3
+ 5 𝑥 2
𝑄 𝑥, 𝑦 = 𝑥2
+ 2𝑥𝑦 + 𝑦2
𝑥, 𝑦 3
Según la cantidad de términos se conoce lo siguiente:
N° DE TÉRMINOS NOMBRE
1 Monomio
2 Binomio
3 Trinomio
Polinomio:
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:
Encuentre la cantidad de valores que toma n,
si la expresión
𝑃 𝑥 = 3𝑥
𝑛−1
2 + 7𝑥8−𝑛 + 5
es un polinomio.
Resolución:
Como P(x) es un polinomio, entonces los exponentes
de su variable son enteros no negativos.
𝑛 − 1
2
𝑒𝑠 𝑒𝑛𝑡𝑒𝑟𝑜 𝑛𝑜 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑜
𝑒𝑛𝑡𝑜𝑛𝑐𝑒𝑠 𝑛 = 1; 3; 5; 7; 9; 11; …
Además:
8 − 𝑛 𝑒𝑠 𝑒𝑛𝑡𝑒𝑟𝑜 𝑛𝑜 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑜
𝑒𝑛𝑡𝑜𝑛𝑐𝑒𝑠 𝑛 = 8; 7; 6; 5; 4; …
De ambos casos, se concluye que los valores comunes
son
𝑛 = 1; 3; 5; 7
Entonces n toma 4 valores.
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
VALOR NUMÉRICO (VN)
El valor numérico de una expresión es el resultado que queda al evaluar dicha expresión, cuando
su(s) variable(s) toma(n) valor(es) fijo(s).
Ejemplos
𝑃 𝑥 = 2𝑥 + 3
𝑆𝑖 𝑥 = 5 𝑃 5 = 2 5 + 3 = 13
𝑆𝑖 𝑥 = −4 𝑃 −4 = 2 −4 + 3 = −5
𝑆𝑖 𝑥 =
5
2
𝑃
5
2
= 2
5
2
+ 3 = 8
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
POLINOMIOS DE UNA VARIABLE
La forma general de un polinomio de una variable es:
𝑃 𝑥 = 𝑎0𝑥𝑛
+ 𝑎1𝑥𝑛−1
+ 𝑎2𝑥𝑛−2
+ ⋯ + 𝑎𝑛−1𝑥 + 𝑎𝑛
Donde:
Variable: x
Coeficientes: 𝑎0, 𝑎1, 𝑎2, … , 𝑎𝑛
°[P]=n: Es el grado de un polinomio, y se define como
el mayor exponente que tiene su variable.
Ejemplo:
𝑃 𝑥 = 4𝑥5
+ 8𝑥3
− 7𝑥9
− 2
Variable: x
Coeficientes: 4; 8; -7; -2
°[P] =9, el mayor exponente de la variable x es 9.
NOTA:
El coeficiente principal de un polinomio es el coeficiente
del término de mayor exponente. Ejemplo:
𝑁 𝑥 = 4 − 3𝑥2
+ 7𝑥4
− 8𝑥3
− 2𝑥
El término de mayor exponente es 7𝑥4
El coeficiente principal es 7
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
Polinomio cúbico: Es aquel polinomio de grado 3. Su forma general es:
𝑃 𝑥 = 𝑎𝑥3 + 𝑏𝑥2 + 𝑐𝑥 + 𝑑
Ejemplos:
𝑀 𝑥 = 4𝑥3
− 7𝑥2
+ 5𝑥 −
1
2
Término independiente
Término lineal
Término cuadrático
Término cúbico
𝑁 𝑥 =
3
2
𝑥2
− 2𝑥3
Término cúbico 4𝑥3
Término cuadrático −7𝑥2
Término lineal 5𝑥
Término
independiente −
1
2
Término cúbico − 2𝑥3
Término cuadrático
3
2
𝑥2
Término lineal No tiene
Término independiente No tiene
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
Polinomio cuadrático: Es aquel polinomio de grado 2. Su forma general es:
𝑃 𝑥 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 Término independiente
Término lineal
Término cuadrático
Ejemplos:
𝑅 𝑥 = −
2
3
𝑥2
+ 5𝑥 − 7 𝑇 𝑥 = 8 − 4𝑥2
Término cuadrático −
2
3
𝑥2
Término lineal 5𝑥
Término independiente − 7
Término cuadrático −4𝑥2
Término lineal 𝑁𝑜 𝑡𝑖𝑒𝑛𝑒
Término independiente 8
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
Polinomio lineal: Su forma general es:
Es aquel polinomio de grado 1.
𝑃 𝑥 = 𝑎𝑥 + 𝑏 Término independiente
Término lineal
Ejemplos:
𝑀 𝑥 = 8𝑥 + 7 𝑁 𝑥 =
2
3
𝑥
Término lineal 8𝑥
Término independiente 7
Término lineal
2
3
𝑥
Término independiente No tiene
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
PROPIEDADES:
Si tenemos el polinomio: 𝑃 𝑥 = 𝑎0𝑥𝑛
+ 𝑎1𝑥𝑛−1
+ 𝑎2𝑥𝑛−2
+ ⋯ + 𝑎𝑛−1𝑥 + 𝑎𝑛
Entonces se cumple:
𝑃 1 = 𝑎0 + 𝑎1 + 𝑎2 + ⋯ + 𝑎𝑛−1 + 𝑎𝑛 = 𝑠𝑢𝑚𝑎 𝑑𝑒 𝑐𝑜𝑒𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑒𝑠 𝑑𝑒 𝑃 𝑥
Además:
𝑃 0 = 𝑎𝑛 = 𝑡é𝑟𝑚𝑖𝑛𝑜 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑖𝑒𝑛𝑡𝑒
Ejemplo:
Dado el polinomio 𝑃 𝑥 = 2𝑥 + 1 4
+ 𝑥 + 1 5
− 10
encuentre el término independiente y la suma de
coeficientes del polinomio.
Resolución: Término independiente:
𝑃 0
𝑃 0
Suma de coeficientes:
𝑃 1
𝑃 1
+ 0 + 1 5
= 2 0 + 1 4 −10
= −8
= 14 + 15 − 10
= 2 1 + 1 4
−10
+ 1 + 1 5
= 34
+ 25
− 10 = 103
12. 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