Lecture 5 - Polynomials | Algebra- Engineering Maths - Engineering Mathematics PDF Download

 Page 1


Polynomials 
Institute of Lifelong Learning                                                                                    1 
 
 
 
 
 
 
 
 
Lesson: Polynomials 
Course Developer: Dr Binay Kumar Sharma 
Affiliation: S.B.S. College, University of Delhi 
 
 
 
 
  
Page 2


Polynomials 
Institute of Lifelong Learning                                                                                    1 
 
 
 
 
 
 
 
 
Lesson: Polynomials 
Course Developer: Dr Binay Kumar Sharma 
Affiliation: S.B.S. College, University of Delhi 
 
 
 
 
  
Polynomials 
Institute of Lifelong Learning                                                                                    2 
Table of Contents 
 Chapter : Polynomials 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: Polynomials 
? 3.1: Zero of a polynomial 
? 3.2: Algebra of Polynomials 
? 4: Algebraic Structure 
? 4.1: Ring 
? 4.2: Commutative Ring 
? 4.3: Commutative ring with unity 
? 4.4: Zero Divisors 
? 4.5: Integral Domain 
? 4.6: Field 
? 5: Fundamental theorem of algebra 
? 5.1: Geometrical Interpretation of the solution of 
polynomial equation 
? 6: Factor theorem 
? 7: Irreducible Polynomial 
? 8: Multiplicity of Zero 
? 9: Consequences of Fundamental Theorem of Algebra 
? 10: Descartes' Rule of Signs 
? 11: Relation between roots and coefficients of a polynomial 
? 12: Bounds on the Zeroes of a Polynomial 
? Summary  
? Exercises 
? Glossary 
? References 
 
Page 3


Polynomials 
Institute of Lifelong Learning                                                                                    1 
 
 
 
 
 
 
 
 
Lesson: Polynomials 
Course Developer: Dr Binay Kumar Sharma 
Affiliation: S.B.S. College, University of Delhi 
 
 
 
 
  
Polynomials 
Institute of Lifelong Learning                                                                                    2 
Table of Contents 
 Chapter : Polynomials 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: Polynomials 
? 3.1: Zero of a polynomial 
? 3.2: Algebra of Polynomials 
? 4: Algebraic Structure 
? 4.1: Ring 
? 4.2: Commutative Ring 
? 4.3: Commutative ring with unity 
? 4.4: Zero Divisors 
? 4.5: Integral Domain 
? 4.6: Field 
? 5: Fundamental theorem of algebra 
? 5.1: Geometrical Interpretation of the solution of 
polynomial equation 
? 6: Factor theorem 
? 7: Irreducible Polynomial 
? 8: Multiplicity of Zero 
? 9: Consequences of Fundamental Theorem of Algebra 
? 10: Descartes' Rule of Signs 
? 11: Relation between roots and coefficients of a polynomial 
? 12: Bounds on the Zeroes of a Polynomial 
? Summary  
? Exercises 
? Glossary 
? References 
 
Polynomials 
Institute of Lifelong Learning                                                                                    3 
1. Learning Outcomes 
After you have read this chapter, you should be able to  
? define the Polynomial, 
? degree of a polynomial, 
? Algebra of polynomials, 
? Zero of Polynomial, 
? Multiplicity of zero,  
? Algebraic Structures. 
? You should be able to know how many zeroes of a Polynomial will 
have, 
? how we can factorize the polynomial, irreducibility of a Polynomial. 
? Descartes' rule of Signs will help you to find how many positive roots; 
negative roots will have of a polynomial equation. 
2. Introduction 
In this chapter we shall study first of all the concept of polynomial which 
plays a central role in the Mathematics and in basic science. The concept of 
polynomial is one of the most important concepts of Mathematics. Every 
Mathematical Model of any situation in any subject, involves some kind of an 
equation. Polynomial equations are most used equations. The fundamental 
theorem of Algebra is a theorem about equation solving. This theorem says 
that every polynomial equation with complex coefficients has at least one 
complex solution. With the help of Descartes' rule Signs we can discuss the 
stability of the periodic orbit of a satellite.  
 
3. Polynomial 
 A function of a single variable is said to be Polynomial on its domain if 
we can put it in the form 
 
? ?
1 1 0
1 1 0
....
nn
nn
p t a t a t at a t
?
?
? ? ? ? ?
   (1)
 
Page 4


Polynomials 
Institute of Lifelong Learning                                                                                    1 
 
 
 
 
 
 
 
 
Lesson: Polynomials 
Course Developer: Dr Binay Kumar Sharma 
Affiliation: S.B.S. College, University of Delhi 
 
 
 
 
  
Polynomials 
Institute of Lifelong Learning                                                                                    2 
Table of Contents 
 Chapter : Polynomials 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: Polynomials 
? 3.1: Zero of a polynomial 
? 3.2: Algebra of Polynomials 
? 4: Algebraic Structure 
? 4.1: Ring 
? 4.2: Commutative Ring 
? 4.3: Commutative ring with unity 
? 4.4: Zero Divisors 
? 4.5: Integral Domain 
? 4.6: Field 
? 5: Fundamental theorem of algebra 
? 5.1: Geometrical Interpretation of the solution of 
polynomial equation 
? 6: Factor theorem 
? 7: Irreducible Polynomial 
? 8: Multiplicity of Zero 
? 9: Consequences of Fundamental Theorem of Algebra 
? 10: Descartes' Rule of Signs 
? 11: Relation between roots and coefficients of a polynomial 
? 12: Bounds on the Zeroes of a Polynomial 
? Summary  
? Exercises 
? Glossary 
? References 
 
Polynomials 
Institute of Lifelong Learning                                                                                    3 
1. Learning Outcomes 
After you have read this chapter, you should be able to  
? define the Polynomial, 
? degree of a polynomial, 
? Algebra of polynomials, 
? Zero of Polynomial, 
? Multiplicity of zero,  
? Algebraic Structures. 
? You should be able to know how many zeroes of a Polynomial will 
have, 
? how we can factorize the polynomial, irreducibility of a Polynomial. 
? Descartes' rule of Signs will help you to find how many positive roots; 
negative roots will have of a polynomial equation. 
2. Introduction 
In this chapter we shall study first of all the concept of polynomial which 
plays a central role in the Mathematics and in basic science. The concept of 
polynomial is one of the most important concepts of Mathematics. Every 
Mathematical Model of any situation in any subject, involves some kind of an 
equation. Polynomial equations are most used equations. The fundamental 
theorem of Algebra is a theorem about equation solving. This theorem says 
that every polynomial equation with complex coefficients has at least one 
complex solution. With the help of Descartes' rule Signs we can discuss the 
stability of the periodic orbit of a satellite.  
 
3. Polynomial 
 A function of a single variable is said to be Polynomial on its domain if 
we can put it in the form 
 
? ?
1 1 0
1 1 0
....
nn
nn
p t a t a t at a t
?
?
? ? ? ? ?
   (1)
 
Polynomials 
Institute of Lifelong Learning                                                                                    4 
where 
1 2 1 0
, , ... ,
n n n
a a a a a
??
 
are constant ( real or complex). From the definition 
of Polynomial it is clear that every polynomial can be expressed as a finite 
sum of monomial terms of the form 
k
k
at in which the variable is raised to a 
non-negative integer power. In the polynomial ? ? 1, with 0
n
a ?
 
the numbers 
? ? 0
i
a i n ?? are called coefficients, 
n
a is the leading coefficient, 
n
n
at the 
leading term, 
0
a
 
is the constant term or constant coefficient, 
1
a
 
is the linear 
coefficient and 
1
at
 
the linear term. When the leading coefficient 
n
a
 
is 1 the 
polynomial is called the Monic. The non-negative integer n is called the 
degree of the polynomial, written as ? ? degpn ? . A non-zero constant 
polynomial has degree 0. 
Example 1: 
 
? ?
6 5 2 1
22
8 2 1 1,
cos2 cos 2cos 1 2 1,
cos
t t t t
arc t t
wheret
?
? ? ? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ?
 
are polynomials of degree 6 and 2 respectively but 
1
3
t and sint are not 
polynomials.  
Value Additions: Special names of low degree polynomials 
 
S. No. Degree of the 
Polynomial 
Type of the Polynomial 
1 1 Linear 
2 2 Quadratic 
3 3 Cubic 
4 4 Quartic 
5 5 Quantic 
 
 
Page 5


Polynomials 
Institute of Lifelong Learning                                                                                    1 
 
 
 
 
 
 
 
 
Lesson: Polynomials 
Course Developer: Dr Binay Kumar Sharma 
Affiliation: S.B.S. College, University of Delhi 
 
 
 
 
  
Polynomials 
Institute of Lifelong Learning                                                                                    2 
Table of Contents 
 Chapter : Polynomials 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: Polynomials 
? 3.1: Zero of a polynomial 
? 3.2: Algebra of Polynomials 
? 4: Algebraic Structure 
? 4.1: Ring 
? 4.2: Commutative Ring 
? 4.3: Commutative ring with unity 
? 4.4: Zero Divisors 
? 4.5: Integral Domain 
? 4.6: Field 
? 5: Fundamental theorem of algebra 
? 5.1: Geometrical Interpretation of the solution of 
polynomial equation 
? 6: Factor theorem 
? 7: Irreducible Polynomial 
? 8: Multiplicity of Zero 
? 9: Consequences of Fundamental Theorem of Algebra 
? 10: Descartes' Rule of Signs 
? 11: Relation between roots and coefficients of a polynomial 
? 12: Bounds on the Zeroes of a Polynomial 
? Summary  
? Exercises 
? Glossary 
? References 
 
Polynomials 
Institute of Lifelong Learning                                                                                    3 
1. Learning Outcomes 
After you have read this chapter, you should be able to  
? define the Polynomial, 
? degree of a polynomial, 
? Algebra of polynomials, 
? Zero of Polynomial, 
? Multiplicity of zero,  
? Algebraic Structures. 
? You should be able to know how many zeroes of a Polynomial will 
have, 
? how we can factorize the polynomial, irreducibility of a Polynomial. 
? Descartes' rule of Signs will help you to find how many positive roots; 
negative roots will have of a polynomial equation. 
2. Introduction 
In this chapter we shall study first of all the concept of polynomial which 
plays a central role in the Mathematics and in basic science. The concept of 
polynomial is one of the most important concepts of Mathematics. Every 
Mathematical Model of any situation in any subject, involves some kind of an 
equation. Polynomial equations are most used equations. The fundamental 
theorem of Algebra is a theorem about equation solving. This theorem says 
that every polynomial equation with complex coefficients has at least one 
complex solution. With the help of Descartes' rule Signs we can discuss the 
stability of the periodic orbit of a satellite.  
 
3. Polynomial 
 A function of a single variable is said to be Polynomial on its domain if 
we can put it in the form 
 
? ?
1 1 0
1 1 0
....
nn
nn
p t a t a t at a t
?
?
? ? ? ? ?
   (1)
 
Polynomials 
Institute of Lifelong Learning                                                                                    4 
where 
1 2 1 0
, , ... ,
n n n
a a a a a
??
 
are constant ( real or complex). From the definition 
of Polynomial it is clear that every polynomial can be expressed as a finite 
sum of monomial terms of the form 
k
k
at in which the variable is raised to a 
non-negative integer power. In the polynomial ? ? 1, with 0
n
a ?
 
the numbers 
? ? 0
i
a i n ?? are called coefficients, 
n
a is the leading coefficient, 
n
n
at the 
leading term, 
0
a
 
is the constant term or constant coefficient, 
1
a
 
is the linear 
coefficient and 
1
at
 
the linear term. When the leading coefficient 
n
a
 
is 1 the 
polynomial is called the Monic. The non-negative integer n is called the 
degree of the polynomial, written as ? ? degpn ? . A non-zero constant 
polynomial has degree 0. 
Example 1: 
 
? ?
6 5 2 1
22
8 2 1 1,
cos2 cos 2cos 1 2 1,
cos
t t t t
arc t t
wheret
?
? ? ? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ?
 
are polynomials of degree 6 and 2 respectively but 
1
3
t and sint are not 
polynomials.  
Value Additions: Special names of low degree polynomials 
 
S. No. Degree of the 
Polynomial 
Type of the Polynomial 
1 1 Linear 
2 2 Quadratic 
3 3 Cubic 
4 4 Quartic 
5 5 Quantic 
 
 
Polynomials 
Institute of Lifelong Learning                                                                                    5 
3.1. Zero of a polynomial: A number r is said to be a zero of a polynomial 
? ? pt iff ? ? 0 pr ? when  ? ? 0 pr ? , we say that r is a root or a solution of the 
polynomial equation ? ? 0 pt ? . 
3.2. Algebra of Polynomials 
Let   ? ?
1 1 0
1 1 0
....
nn
nn
p t a t a t at a t
?
?
? ? ? ? ? and 
 
? ?
1 1 0
1 1 0
....
nn
nn
q t bt b t bt bt
?
?
? ? ? ? ? 
be two polynomials then  
(i) Sum of two polynomials 
 
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
1 1 0
1 1 1 1 0 0
....
nn
n n n n
p t q t p q t a b t a b t a b t a b t
?
??
? ? ? ? ? ? ? ? ? ? ? ? 
(ii) Difference of two polynomials 
 
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
1 1 0
1 1 1 1 0 0
....
nn
n n n n
p t q t p q t a b t a b t a b t a b t
?
??
? ? ? ? ? ? ? ? ? ? ? ? 
(iii) Product of a constant and a polynomial  
 
? ? ? ?
1 1 0
1 1 0
....
nn
nn
cp t ca t ca t cat ca t
?
?
? ? ? ? ? 
(iv) Product of two polynomials  
 
? ? ? ? ? ? ? ? ? ? ? ?
? ?
0 0 0 1 1 0 0 1 1 0
... ... ...
...
r r i r i r
mn
nm
pq t p t q t a b a b ab t a b ab ab a b t
a b t
??
?
? ? ? ? ? ? ? ? ? ? ?
??
 
(v) Composition of two polynomials 
 
? ? ? ? ? ? ? ?
pq t p q t ? .  
4. Algebraic Structure 
4.1.Ring: 
A Ring ,, R ??
 
is a set R
 
together with two binary operations '' ?
 
and '' ?
 
called 
addition and multiplication respectively defined on R
 
such that the following 
axioms are satisfied: 
(i) Closure Property 
 
(ii) Commutative Property  
 
, a b b a a b R ? ? ? ? ? 
, a b S a b R ? ? ? ?