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
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
This PPT explains the concept of polynomial in detail. It describes the meaning of polynomials with the help of different examples.Furthermore different types of polynomials on the basis of degree and number of terms.This will be helpful for students and for teachers.
This PPT explains the concept of polynomial in detail. It describes the meaning of polynomials with the help of different examples.Furthermore different types of polynomials on the basis of degree and number of terms.This will be helpful for students and for teachers.
Power Point Presentation on a PAIR OF LINEAR EQUATION IN TWO VARIABLES, MATHS project...
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This PPT will clarify your all doubts in Arithmetic Progression.
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Fashionista Chic Couture Maze & Coloring Adventures is a coloring and activity book filled with many maze games and coloring activities designed to delight and engage young fashion enthusiasts. Each page offers a unique blend of fashion-themed mazes and stylish illustrations to color, inspiring creativity and problem-solving skills in children.
This document announces the winners of the 2024 Youth Poster Contest organized by MATFORCE. It lists the grand prize and age category winners for grades K-6, 7-12, and individual age groups from 5 years old to 18 years old.
Hadj Ounis's most notable work is his sculpture titled "Metamorphosis." This piece showcases Ounis's mastery of form and texture, as he seamlessly combines metal and wood to create a dynamic and visually striking composition. The juxtaposition of the two materials creates a sense of tension and harmony, inviting viewers to contemplate the relationship between nature and industry.
Boudoir photography, a genre that captures intimate and sensual images of individuals, has experienced significant transformation over the years, particularly in New York City (NYC). Known for its diversity and vibrant arts scene, NYC has been a hub for the evolution of various art forms, including boudoir photography. This article delves into the historical background, cultural significance, technological advancements, and the contemporary landscape of boudoir photography in NYC.
This tutorial offers a step-by-step guide on how to effectively use Pinterest. It covers the basics such as account creation and navigation, as well as advanced techniques including creating eye-catching pins and optimizing your profile. The tutorial also explores collaboration and networking on the platform. With visual illustrations and clear instructions, this tutorial will equip you with the skills to navigate Pinterest confidently and achieve your goals.
2. Introduction
An algebraic expression in which variables involved have only
non-negative integral powers is called a polynomial.
E.g.- (a) 2x3
–4x2
+6x–3 is a polynomial in one variable x.
(b) 8p7+4p2+11p3-9p is a polynomial in one variable p.
(c) 4+7x4/5
+9x5 is an expression but not a polynomial
since it contains a term x4/5
, where 4/5
is not
a non-negative integer.
3. Degree of a Polynomial in one
variable.
• What is degree of the following binomial?
35 2
+x
The answer is 2. 5x2
+ 3 is a polynomial in x of degree 2.
In case of a polynomial in one variable, the highest power of the
variable is called the degree of polynomial.
4. Degree of a Polynomial in two
variables.
• What is degree of the following polynomial?
49375 332
+++− yxyxyx
In case of polynomials on more than one variable, the sum of
powers of the variables in each term is taken up and the highest
sum so obtained is called the degree of polynomial.
• The answer is five because if we add 2
and 3
, the answer is five
which is the highest power in the whole polynomial.
E.g.- is a polynomial in x
and y of degree 7.
92853 243
+++− yxyxyx
5. Polynomials in one variable
• A polynomial is a monomial or a sum of monomials.
• Each monomial in a polynomial is a term of the
polynomial.
The number factor of a term is called the coefficient.
The coefficient of the first term in a polynomial is the
lead coefficient.
• A polynomial with two terms is called a binomial.
• A polynomial with three term is called a trinomial.
6. Polynomials in one variable
The degree of a polynomial in one variable is the largest
exponent of that variable.
14 +x
A constant has no variable. It is a 0 degree polynomial.2
This is a 1st degree polynomial. 1st degree polynomials
are linear.
1425 2
−+ xx This is a 2nd degree polynomial. 2nd
degree polynomials are quadratic.
183 3
−x This is a 3rd degree polynomial. 3rd degree
polynomials are cubic.
8. Standard Form
Phase 1Phase 1 Phase 2Phase 2
To rewrite a
polynomial in
standard form,
rearrange the
terms of the
polynomial
starting with the
largest degree
term and ending
with the lowest
degree term.
The leading coefficient,
the coefficient of the
first term in a
polynomial written in
standard form, should
be positive.
How to convert a polynomial into standard form?
9. Remainder Theorem
TEXT TEXT TEXT TEXT
Let f(x) be a polynomial of degree n > 1 and let a be any real number.
When f(x) is divided by (x-a) , then the remainder is f(a).
PROOF Suppose when f(x) is divided by (x-a), the quotient is g(x) and the remainder
is r(x).
Then, degree r(x) < degree (x-a)
degree r(x) < 1 [ therefore, degree (x-a)=1]
degree r(x) = 0
r(x) is constant, equal to r (say)
Thus, when f(x) is divided by (x-a), then the quotient is g9x) and the remainder is r.
Therefore, f(x) = (x-a)*g(x) + r (i)
Putting x=a in (i), we get r = f(a)
Thus, when f(x) is divided by (x-a), then the remainder is f(a).
10. Questions on Remainder Theorem
Q.) Find the remainder when the polynomial
f(x) = x4
+ 2x3
– 3x2
+ x – 1 is divided by (x-2).
A.) x-2 = 0 x=2
By remainder theorem, we know that when f(x) is divided by (x-2),
the remainder is x(2).
Now, f(2) = (24
+ 2*23
– 3*22
+ 2-1)
= (16 + 16 – 12 + 2 – 1) = 21.
Hence, the required remainder is 21.
11. Factor Theorem
Let f(x) be a polynomial of degree n > 1 and let a be
any real number.
(i) If f(a) = 0 then (x-a) is a factor of f(x).
PROOF let f(a) = 0
On dividing f(x) by 9x-a), let g(x) be the quotient. Also,
by remainder theorem, when f(x) is divided by (x-a),
then the remainder is f(a).
therefore f(x) = (x-a)*g(x) + f(a)
f(x) = (x-a)*g(x) [therefore f(a)=0(given]
(x-a) is a factor of f(x).
12. Algebraic Identities
Some common identities used to factorize polynomials
(x+a)(x+b)=x2+(a+b)x+ab(a+b)2
=a2
+b2
+2ab (a-b)2
=a2
+b2
-2ab a2
-b2
=(a+b)(a-b)
14. Q/A on Polynomials
Q.1) Show that (x-3) is a factor of polynomial
f(x)=x3+x2-17x+15.
A.1) By factor theorem, (x-3) will be a factor of f(x) if f(3)=0.
Now, f(x)=x3+x2-17x+15
f(3)=(33+32-17*3+15)=(27+9-51+15)=0
(x-3) is a factor of f(x).
Hence, (x-3) is a factor of the given polynomial f(x).
16. Points to Remember
• A real number ‘a’ is a zero of a polynomial p(x) if p(a)=0. In
this case, a is also called a root of the equation p(x)=0.
• Every linear polynomial in one variable has a unique zero, a
non-zero constant polynomial has no zero, and every real
number is a zero of the zero polynomial.