This document defines key concepts related to polynomials, including:
- Variables and constants, with variables represented by letters like x and y.
- Algebraic expressions that combine constants and variables.
- The coefficient as the number in front of a variable.
- A term as a part of an algebraic expression separated by addition or subtraction.
- The degree as the highest power of the variable in a polynomial.
- A polynomial defined as an algebraic expression with positive whole number exponents, that can be classified based on its degree or number of terms.
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.
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.
All the best to all students of class IX...This PPT will makes your difficulties easy to do....You will understand the polynomial chapter easily by seeing this ....Thanks for watching this ..Please Share, Like and Subscribe the PPT
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All the best to all students of class IX...This PPT will makes your difficulties easy to do....You will understand the polynomial chapter easily by seeing this ....Thanks for watching this ..Please Share, Like and Subscribe the PPT
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.
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http://sandymillin.wordpress.com/iateflwebinar2024
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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.
1. CONSTANT
A value that does not change.
VARIABLE
A letter that represents an unknown number.
Eg: commonly used letters to represent variables are 𝑥, 𝑦, 𝑧, and 𝑡
Prior Knowledge
1
Subtopic- 1
Introduction to polynomials
Learning objectives:
• To recall concepts of variables , constants
and algebraic expressions
• To define polynomial
• To understand the concept of degree
• To classify polynomials on the basis of
degree and number of terms.
3. COEFFICIENT
a number in front of a variable .
Example: 6z means 6 times z
so 6 is a coefficient
Example: x is really 1x.
Prior Knowledge
3
4. TERM
The part of an algebraic expression separated by
addition or subtraction.
Prior Knowledge
4
5. Degree
DEGREE: the highest power of the variable.
Eg: (i). 5x5+4x2+3
The degree is 5.
(ii)12x3 -5x2 + 2
The degree is 3.
(iii) 4x +12
The degree is 1.
5
7. So, let’s define Polynomial Mathematically
A polynomial is an algebraic expression which is a
combination of the constants, variables and positive
exponents(whole number)
Definition
7
8. Now, we can classify polynomial on the basis of terms and degrees
FIRSTLY,CLASSIFICATION OF POLYNOMIALS ON THE BASIS OF TERMS:-
POLYNOMIALS
MONOMIAL eg:3x
(one term)
BINOMIAL eg:
x – y(two terms)
TRINOMIAL. eg:
3x2 − 3y2 + 2.(three
terms)
On theBasis of
terms
8
9. On the Basis of
degrees
CLASSIFICATION OF POLYNOMIALS ON THE BASIS OF DEGREE:-
9
10. Examples
POLYNOMIALS DEGREE CLASSIFY BY
DEGREE
CLASSIFY BY
NO.OF TERMS
5 0 Constant Monomial
2x- 4 1 Linear Binomial
3𝑥3+4x+1 3 Cubic Trinomial
10
11. +.STANDARD FORM
+The Standard Form writing a polynomial is to put the
terms with the highest degree first.
Example: Put this in Standard Form 3𝑥2
- 7 +4𝑥3
+ 𝑥6
The highest degree is 6, so that goes first, then 3, 2 and
then the constant last:
𝑥6
+ 4𝑥3
+ 3𝑥2
− 7
Standard Form
11
12. Standard Form
Examples: WRITE THE FOLLOWING POLYNOMIALS IN
STANDARD FORM:
1) 8y3-4y+y2-3. 2) 3x2-4x3-2x
3) 6-4y 4) 4a+6a2+5a3
5) n3-5n5+8
Answers: 1)8y3+y2-4y-3 2)-4x3+3x2-2x
3)-4y+6 4)5a3 +6a2+4a
5) -5n5+n3+8
12