POLYNOMIALS
LECTURE-I
Poly means ‘many’ and nomial means numerals or terms.
Therefore, based on meaning, a polynomial is defined as
‘An expression that contains many terms separated by mathematical operations.’
Mathematically,
2 3 n
p( x )=a 0+ a1 ( x ) +a 2 ( x ) + a3 ( x ) +...+ an ( x ) where,
a 0 , a 1 , a2 , ..., a n are real coefficients and n is a whole number.
Every part of above expression is a polynomial in itself.
For example a 0 alone is a polynomial.
THINK 1: Mention atleast 5 different expressions which are not polynomials.
A polynomial is uniquely identified by its degree
DEGREE: The highest power of the variable in the given polynomial is called the degree of
the polynomial.
For examples:
a) p( x )=2 x is a polynomial having degree 1 as the power of the variable x here is 1
b) p(x )=2 x2 + x +5 is a polynomial having degree 2 as the power of the variable x here is 2
c) p( y )=a y 3 +b y 2 +cy + d is a polynomial having degree 3 as the power of the variable y
here is 3.
NOTE 1: The coefficients a 0 , a 1 , a2 , ..., a n must be understood in the following sense:
a 0 is the coefficient of the variable having power 0.
a 1 is the coefficient of the variable having power 1.
a 2 is the coefficient of the variable having power 2.
And so on.
QUESTION 1…..What will be the value of all coefficients available in the followings?
a) p( z )=z 6 +2 z 7 b) p(a)=4 a+2 a2 +5 a3
NOTE 2: A polynomial must be understood in the following sense:
Polynomial is written as
p( x )=a 0+ a1 ( x ) +a 2 ( x )2+ a3 ( x )3 +...+ an ( x )n
It means p of x, i.e the relation of p is shown with x and hence is written p( x ).
So, p(1)would mean to substitute 1 in place of x wherever x (the variable) is present.
2 3 n
i.e p(1)=a 0+ a1 ( 1 )+ a2 (1 ) + a3 ( 1 ) +...+a n ( 1 ) ¿ a0 + a1+ a2 +...+a n
QUESTION 2…..Find:
a) p ¿in p(a)=( 2 a )2 b) p( y +2)in p( x )=x 3+ x2
Though a polynomial is a general word for the vast majority of expressions. It’s required to
classify polynomials so that we can study them easily.
What are the different types of polynomials?
TYPES OF POLYNOMIAL:
We classify polynomials based on the degree of the polynomials.
DEGREE: The highest power of the variable in the given polynomial is called the degree of
the polynomial.
i) If the degree is 0, then the polynomial is called a constant polynomial.
LECTURE-I
Poly means ‘many’ and nomial means numerals or terms.
Therefore, based on meaning, a polynomial is defined as
‘An expression that contains many terms separated by mathematical operations.’
Mathematically,
2 3 n
p( x )=a 0+ a1 ( x ) +a 2 ( x ) + a3 ( x ) +...+ an ( x ) where,
a 0 , a 1 , a2 , ..., a n are real coefficients and n is a whole number.
Every part of above expression is a polynomial in itself.
For example a 0 alone is a polynomial.
THINK 1: Mention atleast 5 different expressions which are not polynomials.
A polynomial is uniquely identified by its degree
DEGREE: The highest power of the variable in the given polynomial is called the degree of
the polynomial.
For examples:
a) p( x )=2 x is a polynomial having degree 1 as the power of the variable x here is 1
b) p(x )=2 x2 + x +5 is a polynomial having degree 2 as the power of the variable x here is 2
c) p( y )=a y 3 +b y 2 +cy + d is a polynomial having degree 3 as the power of the variable y
here is 3.
NOTE 1: The coefficients a 0 , a 1 , a2 , ..., a n must be understood in the following sense:
a 0 is the coefficient of the variable having power 0.
a 1 is the coefficient of the variable having power 1.
a 2 is the coefficient of the variable having power 2.
And so on.
QUESTION 1…..What will be the value of all coefficients available in the followings?
a) p( z )=z 6 +2 z 7 b) p(a)=4 a+2 a2 +5 a3
NOTE 2: A polynomial must be understood in the following sense:
Polynomial is written as
p( x )=a 0+ a1 ( x ) +a 2 ( x )2+ a3 ( x )3 +...+ an ( x )n
It means p of x, i.e the relation of p is shown with x and hence is written p( x ).
So, p(1)would mean to substitute 1 in place of x wherever x (the variable) is present.
2 3 n
i.e p(1)=a 0+ a1 ( 1 )+ a2 (1 ) + a3 ( 1 ) +...+a n ( 1 ) ¿ a0 + a1+ a2 +...+a n
QUESTION 2…..Find:
a) p ¿in p(a)=( 2 a )2 b) p( y +2)in p( x )=x 3+ x2
Though a polynomial is a general word for the vast majority of expressions. It’s required to
classify polynomials so that we can study them easily.
What are the different types of polynomials?
TYPES OF POLYNOMIAL:
We classify polynomials based on the degree of the polynomials.
DEGREE: The highest power of the variable in the given polynomial is called the degree of
the polynomial.
i) If the degree is 0, then the polynomial is called a constant polynomial.
