Page 1
28 MATHEMA TICS
CHAPTER 2
POLYNOMIALS
2.1 Introduction
You have studied algebraic expressions, their addition, subtraction, multiplication and
division in earlier classes. You also have studied how to factorise some algebraic
expressions. You may recall the algebraic identities :
(x + y)
2
= x
2
+ 2xy + y
2
(x – y)
2
= x
2
– 2xy + y
2
and x
2
– y
2
= (x + y) (x – y)
and their use in factorisation. In this chapter, we shall start our study with a particular
type of algebraic expression, called polynomial, and the terminology related to it. We
shall also study the Remainder Theorem and Factor Theorem and their use in the
factorisation of polynomials. In addition to the above, we shall study some more algebraic
identities and their use in factorisation and in evaluating some given expressions.
2.2 Polynomials in One V ariable
Let us begin by recalling that a variable is denoted by a symbol that can take any real
value. We use the letters x, y, z, etc. to denote variables. Notice that 2x, 3x, – x, – x
are algebraic expressions. All these expressions are of the form (a constant) × x. Now
suppose we want to write an expression which is (a constant) × (a variable) and we do
not know what the constant is. In such cases, we write the constant as a, b, c, etc. So
the expression will be ax, say.
However, there is a difference between a letter denoting a constant and a letter
denoting a variable. The values of the constants remain the same throughout a particular
situation, that is, the values of the constants do not change in a given problem, but the
value of a variable can keep changing.
2022-23
Page 2
28 MATHEMA TICS
CHAPTER 2
POLYNOMIALS
2.1 Introduction
You have studied algebraic expressions, their addition, subtraction, multiplication and
division in earlier classes. You also have studied how to factorise some algebraic
expressions. You may recall the algebraic identities :
(x + y)
2
= x
2
+ 2xy + y
2
(x – y)
2
= x
2
– 2xy + y
2
and x
2
– y
2
= (x + y) (x – y)
and their use in factorisation. In this chapter, we shall start our study with a particular
type of algebraic expression, called polynomial, and the terminology related to it. We
shall also study the Remainder Theorem and Factor Theorem and their use in the
factorisation of polynomials. In addition to the above, we shall study some more algebraic
identities and their use in factorisation and in evaluating some given expressions.
2.2 Polynomials in One V ariable
Let us begin by recalling that a variable is denoted by a symbol that can take any real
value. We use the letters x, y, z, etc. to denote variables. Notice that 2x, 3x, – x, – x
are algebraic expressions. All these expressions are of the form (a constant) × x. Now
suppose we want to write an expression which is (a constant) × (a variable) and we do
not know what the constant is. In such cases, we write the constant as a, b, c, etc. So
the expression will be ax, say.
However, there is a difference between a letter denoting a constant and a letter
denoting a variable. The values of the constants remain the same throughout a particular
situation, that is, the values of the constants do not change in a given problem, but the
value of a variable can keep changing.
2022-23
POLYNOMIALS 29
Now, consider a square of side 3 units (see Fig. 2.1).
What is its perimeter? Y ou know that the perimeter of a square
is the sum of the lengths of its four sides. Here, each side is
3 units. So, its perimeter is 4 × 3, i.e., 12 units. What will be the
perimeter if each side of the square is 10 units? The perimeter
is 4 × 10, i.e., 40 units. In case the length of each side is x
units (see Fig. 2.2), the perimeter is given by 4x units. So, as
the length of the side varies, the perimeter varies.
Can you find the area of the square PQRS? It is
x × x = x
2
square units. x
2
is an algebraic expression. You are
also familiar with other algebraic expressions like
2x, x
2
+ 2x, x
3
– x
2
+ 4x + 7. Note that, all the algebraic
expressions we have considered so far have only whole
numbers as the exponents of the variable. Expressions of this
form are called polynomials in one variable. In the examples
above, the variable is x. For instance, x
3
– x
2
+ 4x + 7 is a
polynomial in x. Similarly, 3y
2
+ 5y is a polynomial in the
variable y and t
2
+ 4 is a polynomial in the variable t.
In the polynomial x
2
+ 2x, the expressions x
2
and 2x are called the terms of the
polynomial. Similarly, the polynomial 3y
2
+ 5y + 7 has three terms, namely, 3y
2
, 5y and
7. Can you write the terms of the polynomial –x
3
+ 4x
2
+ 7x – 2 ? This polynomial has
4 terms, namely, –x
3
, 4x
2
, 7x and –2.
Each term of a polynomial has a coefficient. So, in –x
3
+ 4x
2
+ 7x – 2, the
coefficient of x
3
is –1, the coefficient of x
2
is 4, the coefficient of x is 7 and –2 is the
coefficient of x
0
(Remember, x
0
= 1). Do you know the coefficient of x in x
2
– x + 7?
It is –1.
2 is also a polynomial. In fact, 2, –5, 7, etc. are examples of constant polynomials.
The constant polynomial 0 is called the zero polynomial. This plays a very important
role in the collection of all polynomials, as you will see in the higher classes.
Now, consider algebraic expressions such as x +
2
3
1
,
3 and . + + x y y
x
Do you
know that you can write x +
1
x
= x + x
–1
? Here, the exponent of the second term, i.e.,
x
–1
is –1, which is not a whole number. So, this algebraic expression is not a polynomial.
Again, 3 x + can be written as
1
2
3 x + . Here the exponent of x is
1
2
, which is
not a whole number. So, is 3 x + a polynomial? No, it is not. What about
3
y + y
2
? It is also not a polynomial (Why?).
Fig. 2.1
Fig. 2.2
3
3 3
3
x
x x
x
S
R
P Q
2022-23
Page 3
28 MATHEMA TICS
CHAPTER 2
POLYNOMIALS
2.1 Introduction
You have studied algebraic expressions, their addition, subtraction, multiplication and
division in earlier classes. You also have studied how to factorise some algebraic
expressions. You may recall the algebraic identities :
(x + y)
2
= x
2
+ 2xy + y
2
(x – y)
2
= x
2
– 2xy + y
2
and x
2
– y
2
= (x + y) (x – y)
and their use in factorisation. In this chapter, we shall start our study with a particular
type of algebraic expression, called polynomial, and the terminology related to it. We
shall also study the Remainder Theorem and Factor Theorem and their use in the
factorisation of polynomials. In addition to the above, we shall study some more algebraic
identities and their use in factorisation and in evaluating some given expressions.
2.2 Polynomials in One V ariable
Let us begin by recalling that a variable is denoted by a symbol that can take any real
value. We use the letters x, y, z, etc. to denote variables. Notice that 2x, 3x, – x, – x
are algebraic expressions. All these expressions are of the form (a constant) × x. Now
suppose we want to write an expression which is (a constant) × (a variable) and we do
not know what the constant is. In such cases, we write the constant as a, b, c, etc. So
the expression will be ax, say.
However, there is a difference between a letter denoting a constant and a letter
denoting a variable. The values of the constants remain the same throughout a particular
situation, that is, the values of the constants do not change in a given problem, but the
value of a variable can keep changing.
2022-23
POLYNOMIALS 29
Now, consider a square of side 3 units (see Fig. 2.1).
What is its perimeter? Y ou know that the perimeter of a square
is the sum of the lengths of its four sides. Here, each side is
3 units. So, its perimeter is 4 × 3, i.e., 12 units. What will be the
perimeter if each side of the square is 10 units? The perimeter
is 4 × 10, i.e., 40 units. In case the length of each side is x
units (see Fig. 2.2), the perimeter is given by 4x units. So, as
the length of the side varies, the perimeter varies.
Can you find the area of the square PQRS? It is
x × x = x
2
square units. x
2
is an algebraic expression. You are
also familiar with other algebraic expressions like
2x, x
2
+ 2x, x
3
– x
2
+ 4x + 7. Note that, all the algebraic
expressions we have considered so far have only whole
numbers as the exponents of the variable. Expressions of this
form are called polynomials in one variable. In the examples
above, the variable is x. For instance, x
3
– x
2
+ 4x + 7 is a
polynomial in x. Similarly, 3y
2
+ 5y is a polynomial in the
variable y and t
2
+ 4 is a polynomial in the variable t.
In the polynomial x
2
+ 2x, the expressions x
2
and 2x are called the terms of the
polynomial. Similarly, the polynomial 3y
2
+ 5y + 7 has three terms, namely, 3y
2
, 5y and
7. Can you write the terms of the polynomial –x
3
+ 4x
2
+ 7x – 2 ? This polynomial has
4 terms, namely, –x
3
, 4x
2
, 7x and –2.
Each term of a polynomial has a coefficient. So, in –x
3
+ 4x
2
+ 7x – 2, the
coefficient of x
3
is –1, the coefficient of x
2
is 4, the coefficient of x is 7 and –2 is the
coefficient of x
0
(Remember, x
0
= 1). Do you know the coefficient of x in x
2
– x + 7?
It is –1.
2 is also a polynomial. In fact, 2, –5, 7, etc. are examples of constant polynomials.
The constant polynomial 0 is called the zero polynomial. This plays a very important
role in the collection of all polynomials, as you will see in the higher classes.
Now, consider algebraic expressions such as x +
2
3
1
,
3 and . + + x y y
x
Do you
know that you can write x +
1
x
= x + x
–1
? Here, the exponent of the second term, i.e.,
x
–1
is –1, which is not a whole number. So, this algebraic expression is not a polynomial.
Again, 3 x + can be written as
1
2
3 x + . Here the exponent of x is
1
2
, which is
not a whole number. So, is 3 x + a polynomial? No, it is not. What about
3
y + y
2
? It is also not a polynomial (Why?).
Fig. 2.1
Fig. 2.2
3
3 3
3
x
x x
x
S
R
P Q
2022-23
30 MATHEMA TICS
If the variable in a polynomial is x, we may denote the polynomial by p(x), or q(x),
or r(x), etc. So, for example, we may write :
p(x) = 2x
2
+ 5x – 3
q(x) = x
3
–1
r(y) = y
3
+ y + 1
s(u) = 2 – u – u
2
+ 6u
5
A polynomial can have any (finite) number of terms. For instance, x
150
+ x
149
+ ...
+ x
2
+ x + 1 is a polynomial with 151 terms.
Consider the polynomials 2x, 2, 5x
3
, –5x
2
, y and u
4
. Do you see that each of these
polynomials has only one term? Polynomials having only one term are called monomials
(‘mono’ means ‘one’).
Now observe each of the following polynomials:
p(x) = x + 1, q(x) = x
2
– x, r(y) = y
9
+ 1, t(u) = u
15
– u
2
How many terms are there in each of these? Each of these polynomials has only
two terms. Polynomials having only two terms are called binomials (‘bi’ means ‘two’).
Similarly, polynomials having only three terms are called trinomials
(‘tri’ means ‘three’). Some examples of trinomials are
p(x) = x + x
2
+ p, q(x) =
2
+ x – x
2
,
r(u) = u + u
2
– 2, t(y) = y
4
+ y + 5.
Now, look at the polynomial p(x) = 3x
7
– 4x
6
+ x + 9. What is the term with the
highest power of x ? It is 3x
7
. The exponent of x in this term is 7. Similarly, in the
polynomial q(y) = 5y
6
– 4y
2
– 6, the term with the highest power of y is 5y
6
and the
exponent of y in this term is 6. We call the highest power of the variable in a polynomial
as the degree of the polynomial. So, the degree of the polynomial 3x
7
– 4x
6
+ x + 9
is 7 and the degree of the polynomial 5y
6
– 4y
2
– 6 is 6. The degree of a non-zero
constant polynomial is zero.
Example 1 : Find the degree of each of the polynomials given below:
(i) x
5
– x
4
+ 3 (ii) 2 – y
2
– y
3
+ 2y
8
(iii) 2
Solution : (i) The highest power of the variable is 5. So, the degree of the polynomial
is 5.
(ii) The highest power of the variable is 8. So, the degree of the polynomial is 8.
(iii)The only term here is 2 which can be written as 2x
0
. So the exponent of x is 0.
Therefore, the degree of the polynomial is 0.
2022-23
Page 4
28 MATHEMA TICS
CHAPTER 2
POLYNOMIALS
2.1 Introduction
You have studied algebraic expressions, their addition, subtraction, multiplication and
division in earlier classes. You also have studied how to factorise some algebraic
expressions. You may recall the algebraic identities :
(x + y)
2
= x
2
+ 2xy + y
2
(x – y)
2
= x
2
– 2xy + y
2
and x
2
– y
2
= (x + y) (x – y)
and their use in factorisation. In this chapter, we shall start our study with a particular
type of algebraic expression, called polynomial, and the terminology related to it. We
shall also study the Remainder Theorem and Factor Theorem and their use in the
factorisation of polynomials. In addition to the above, we shall study some more algebraic
identities and their use in factorisation and in evaluating some given expressions.
2.2 Polynomials in One V ariable
Let us begin by recalling that a variable is denoted by a symbol that can take any real
value. We use the letters x, y, z, etc. to denote variables. Notice that 2x, 3x, – x, – x
are algebraic expressions. All these expressions are of the form (a constant) × x. Now
suppose we want to write an expression which is (a constant) × (a variable) and we do
not know what the constant is. In such cases, we write the constant as a, b, c, etc. So
the expression will be ax, say.
However, there is a difference between a letter denoting a constant and a letter
denoting a variable. The values of the constants remain the same throughout a particular
situation, that is, the values of the constants do not change in a given problem, but the
value of a variable can keep changing.
2022-23
POLYNOMIALS 29
Now, consider a square of side 3 units (see Fig. 2.1).
What is its perimeter? Y ou know that the perimeter of a square
is the sum of the lengths of its four sides. Here, each side is
3 units. So, its perimeter is 4 × 3, i.e., 12 units. What will be the
perimeter if each side of the square is 10 units? The perimeter
is 4 × 10, i.e., 40 units. In case the length of each side is x
units (see Fig. 2.2), the perimeter is given by 4x units. So, as
the length of the side varies, the perimeter varies.
Can you find the area of the square PQRS? It is
x × x = x
2
square units. x
2
is an algebraic expression. You are
also familiar with other algebraic expressions like
2x, x
2
+ 2x, x
3
– x
2
+ 4x + 7. Note that, all the algebraic
expressions we have considered so far have only whole
numbers as the exponents of the variable. Expressions of this
form are called polynomials in one variable. In the examples
above, the variable is x. For instance, x
3
– x
2
+ 4x + 7 is a
polynomial in x. Similarly, 3y
2
+ 5y is a polynomial in the
variable y and t
2
+ 4 is a polynomial in the variable t.
In the polynomial x
2
+ 2x, the expressions x
2
and 2x are called the terms of the
polynomial. Similarly, the polynomial 3y
2
+ 5y + 7 has three terms, namely, 3y
2
, 5y and
7. Can you write the terms of the polynomial –x
3
+ 4x
2
+ 7x – 2 ? This polynomial has
4 terms, namely, –x
3
, 4x
2
, 7x and –2.
Each term of a polynomial has a coefficient. So, in –x
3
+ 4x
2
+ 7x – 2, the
coefficient of x
3
is –1, the coefficient of x
2
is 4, the coefficient of x is 7 and –2 is the
coefficient of x
0
(Remember, x
0
= 1). Do you know the coefficient of x in x
2
– x + 7?
It is –1.
2 is also a polynomial. In fact, 2, –5, 7, etc. are examples of constant polynomials.
The constant polynomial 0 is called the zero polynomial. This plays a very important
role in the collection of all polynomials, as you will see in the higher classes.
Now, consider algebraic expressions such as x +
2
3
1
,
3 and . + + x y y
x
Do you
know that you can write x +
1
x
= x + x
–1
? Here, the exponent of the second term, i.e.,
x
–1
is –1, which is not a whole number. So, this algebraic expression is not a polynomial.
Again, 3 x + can be written as
1
2
3 x + . Here the exponent of x is
1
2
, which is
not a whole number. So, is 3 x + a polynomial? No, it is not. What about
3
y + y
2
? It is also not a polynomial (Why?).
Fig. 2.1
Fig. 2.2
3
3 3
3
x
x x
x
S
R
P Q
2022-23
30 MATHEMA TICS
If the variable in a polynomial is x, we may denote the polynomial by p(x), or q(x),
or r(x), etc. So, for example, we may write :
p(x) = 2x
2
+ 5x – 3
q(x) = x
3
–1
r(y) = y
3
+ y + 1
s(u) = 2 – u – u
2
+ 6u
5
A polynomial can have any (finite) number of terms. For instance, x
150
+ x
149
+ ...
+ x
2
+ x + 1 is a polynomial with 151 terms.
Consider the polynomials 2x, 2, 5x
3
, –5x
2
, y and u
4
. Do you see that each of these
polynomials has only one term? Polynomials having only one term are called monomials
(‘mono’ means ‘one’).
Now observe each of the following polynomials:
p(x) = x + 1, q(x) = x
2
– x, r(y) = y
9
+ 1, t(u) = u
15
– u
2
How many terms are there in each of these? Each of these polynomials has only
two terms. Polynomials having only two terms are called binomials (‘bi’ means ‘two’).
Similarly, polynomials having only three terms are called trinomials
(‘tri’ means ‘three’). Some examples of trinomials are
p(x) = x + x
2
+ p, q(x) =
2
+ x – x
2
,
r(u) = u + u
2
– 2, t(y) = y
4
+ y + 5.
Now, look at the polynomial p(x) = 3x
7
– 4x
6
+ x + 9. What is the term with the
highest power of x ? It is 3x
7
. The exponent of x in this term is 7. Similarly, in the
polynomial q(y) = 5y
6
– 4y
2
– 6, the term with the highest power of y is 5y
6
and the
exponent of y in this term is 6. We call the highest power of the variable in a polynomial
as the degree of the polynomial. So, the degree of the polynomial 3x
7
– 4x
6
+ x + 9
is 7 and the degree of the polynomial 5y
6
– 4y
2
– 6 is 6. The degree of a non-zero
constant polynomial is zero.
Example 1 : Find the degree of each of the polynomials given below:
(i) x
5
– x
4
+ 3 (ii) 2 – y
2
– y
3
+ 2y
8
(iii) 2
Solution : (i) The highest power of the variable is 5. So, the degree of the polynomial
is 5.
(ii) The highest power of the variable is 8. So, the degree of the polynomial is 8.
(iii)The only term here is 2 which can be written as 2x
0
. So the exponent of x is 0.
Therefore, the degree of the polynomial is 0.
2022-23
POLYNOMIALS 31
Now observe the polynomials p(x) = 4x + 5, q(y) = 2y, r(t) = t + 2 and
s(u) = 3 – u. Do you see anything common among all of them? The degree of each of
these polynomials is one. A polynomial of degree one is called a linear polynomial.
Some more linear polynomials in one variable are 2x – 1, 2 y + 1, 2 – u. Now, try and
find a linear polynomial in x with 3 terms? You would not be able to find it because a
linear polynomial in x can have at most two terms. So, any linear polynomial in x will
be of the form ax + b, where a and b are constants and a ? 0 (why?). Similarly,
ay + b is a linear polynomial in y.
Now consider the polynomials :
2x
2
+ 5, 5x
2
+ 3x + p, x
2
and x
2
+
2
5
x
Do you agree that they are all of degree two? A polynomial of degree two is called
a quadratic polynomial. Some examples of a quadratic polynomial are 5 – y
2
,
4y + 5y
2
and 6 – y – y
2
. Can you write a quadratic polynomial in one variable with four
different terms? Y ou will find that a quadratic polynomial in one variable will have at
most 3 terms. If you list a few more quadratic polynomials, you will find that any
quadratic polynomial in x is of the form ax
2
+ bx + c, where a ? 0 and a, b, c are
constants. Similarly, quadratic polynomial in y will be of the form ay
2
+ by + c, provided
a ? 0 and a, b, c are constants.
We call a polynomial of degree three a cubic polynomial. Some examples of a
cubic polynomial in x are 4x
3
, 2x
3
+ 1, 5x
3
+ x
2
, 6x
3
– x, 6 – x
3
, 2x
3
+ 4x
2
+ 6x + 7. How
many terms do you think a cubic polynomial in one variable can have? It can have at
most 4 terms. These may be written in the form ax
3
+ bx
2
+ cx + d, where a ? 0 and
a, b, c and d are constants.
Now, that you have seen what a polynomial of degree 1, degree 2, or degree 3
looks like, can you write down a polynomial in one variable of degree n for any natural
number n? A polynomial in one variable x of degree n is an expression of the form
a
n
x
n
+ a
n–1
x
n–1
+ . . . + a
1
x + a
0
where a
0
, a
1
, a
2
, . . ., a
n
are constants and a
n
? 0.
In particular, if a
0
= a
1
= a
2
= a
3
= . . . = a
n
= 0 (all the constants are zero), we get
the zero polynomial, which is denoted by 0. What is the degree of the zero polynomial?
The degree of the zero polynomial is not defined.
So far we have dealt with polynomials in one variable only. We can also have
polynomials in more than one variable. For example, x
2
+ y
2
+ xyz (where variables
are x, y and z) is a polynomial in three variables. Similarly p
2
+ q
10
+ r (where the
variables are p, q and r), u
3
+ v
2
(where the variables are u and v) are polynomials in
three and two variables, respectively. Y ou will be studying such polynomials in detail
later.
2022-23
Page 5
28 MATHEMA TICS
CHAPTER 2
POLYNOMIALS
2.1 Introduction
You have studied algebraic expressions, their addition, subtraction, multiplication and
division in earlier classes. You also have studied how to factorise some algebraic
expressions. You may recall the algebraic identities :
(x + y)
2
= x
2
+ 2xy + y
2
(x – y)
2
= x
2
– 2xy + y
2
and x
2
– y
2
= (x + y) (x – y)
and their use in factorisation. In this chapter, we shall start our study with a particular
type of algebraic expression, called polynomial, and the terminology related to it. We
shall also study the Remainder Theorem and Factor Theorem and their use in the
factorisation of polynomials. In addition to the above, we shall study some more algebraic
identities and their use in factorisation and in evaluating some given expressions.
2.2 Polynomials in One V ariable
Let us begin by recalling that a variable is denoted by a symbol that can take any real
value. We use the letters x, y, z, etc. to denote variables. Notice that 2x, 3x, – x, – x
are algebraic expressions. All these expressions are of the form (a constant) × x. Now
suppose we want to write an expression which is (a constant) × (a variable) and we do
not know what the constant is. In such cases, we write the constant as a, b, c, etc. So
the expression will be ax, say.
However, there is a difference between a letter denoting a constant and a letter
denoting a variable. The values of the constants remain the same throughout a particular
situation, that is, the values of the constants do not change in a given problem, but the
value of a variable can keep changing.
2022-23
POLYNOMIALS 29
Now, consider a square of side 3 units (see Fig. 2.1).
What is its perimeter? Y ou know that the perimeter of a square
is the sum of the lengths of its four sides. Here, each side is
3 units. So, its perimeter is 4 × 3, i.e., 12 units. What will be the
perimeter if each side of the square is 10 units? The perimeter
is 4 × 10, i.e., 40 units. In case the length of each side is x
units (see Fig. 2.2), the perimeter is given by 4x units. So, as
the length of the side varies, the perimeter varies.
Can you find the area of the square PQRS? It is
x × x = x
2
square units. x
2
is an algebraic expression. You are
also familiar with other algebraic expressions like
2x, x
2
+ 2x, x
3
– x
2
+ 4x + 7. Note that, all the algebraic
expressions we have considered so far have only whole
numbers as the exponents of the variable. Expressions of this
form are called polynomials in one variable. In the examples
above, the variable is x. For instance, x
3
– x
2
+ 4x + 7 is a
polynomial in x. Similarly, 3y
2
+ 5y is a polynomial in the
variable y and t
2
+ 4 is a polynomial in the variable t.
In the polynomial x
2
+ 2x, the expressions x
2
and 2x are called the terms of the
polynomial. Similarly, the polynomial 3y
2
+ 5y + 7 has three terms, namely, 3y
2
, 5y and
7. Can you write the terms of the polynomial –x
3
+ 4x
2
+ 7x – 2 ? This polynomial has
4 terms, namely, –x
3
, 4x
2
, 7x and –2.
Each term of a polynomial has a coefficient. So, in –x
3
+ 4x
2
+ 7x – 2, the
coefficient of x
3
is –1, the coefficient of x
2
is 4, the coefficient of x is 7 and –2 is the
coefficient of x
0
(Remember, x
0
= 1). Do you know the coefficient of x in x
2
– x + 7?
It is –1.
2 is also a polynomial. In fact, 2, –5, 7, etc. are examples of constant polynomials.
The constant polynomial 0 is called the zero polynomial. This plays a very important
role in the collection of all polynomials, as you will see in the higher classes.
Now, consider algebraic expressions such as x +
2
3
1
,
3 and . + + x y y
x
Do you
know that you can write x +
1
x
= x + x
–1
? Here, the exponent of the second term, i.e.,
x
–1
is –1, which is not a whole number. So, this algebraic expression is not a polynomial.
Again, 3 x + can be written as
1
2
3 x + . Here the exponent of x is
1
2
, which is
not a whole number. So, is 3 x + a polynomial? No, it is not. What about
3
y + y
2
? It is also not a polynomial (Why?).
Fig. 2.1
Fig. 2.2
3
3 3
3
x
x x
x
S
R
P Q
2022-23
30 MATHEMA TICS
If the variable in a polynomial is x, we may denote the polynomial by p(x), or q(x),
or r(x), etc. So, for example, we may write :
p(x) = 2x
2
+ 5x – 3
q(x) = x
3
–1
r(y) = y
3
+ y + 1
s(u) = 2 – u – u
2
+ 6u
5
A polynomial can have any (finite) number of terms. For instance, x
150
+ x
149
+ ...
+ x
2
+ x + 1 is a polynomial with 151 terms.
Consider the polynomials 2x, 2, 5x
3
, –5x
2
, y and u
4
. Do you see that each of these
polynomials has only one term? Polynomials having only one term are called monomials
(‘mono’ means ‘one’).
Now observe each of the following polynomials:
p(x) = x + 1, q(x) = x
2
– x, r(y) = y
9
+ 1, t(u) = u
15
– u
2
How many terms are there in each of these? Each of these polynomials has only
two terms. Polynomials having only two terms are called binomials (‘bi’ means ‘two’).
Similarly, polynomials having only three terms are called trinomials
(‘tri’ means ‘three’). Some examples of trinomials are
p(x) = x + x
2
+ p, q(x) =
2
+ x – x
2
,
r(u) = u + u
2
– 2, t(y) = y
4
+ y + 5.
Now, look at the polynomial p(x) = 3x
7
– 4x
6
+ x + 9. What is the term with the
highest power of x ? It is 3x
7
. The exponent of x in this term is 7. Similarly, in the
polynomial q(y) = 5y
6
– 4y
2
– 6, the term with the highest power of y is 5y
6
and the
exponent of y in this term is 6. We call the highest power of the variable in a polynomial
as the degree of the polynomial. So, the degree of the polynomial 3x
7
– 4x
6
+ x + 9
is 7 and the degree of the polynomial 5y
6
– 4y
2
– 6 is 6. The degree of a non-zero
constant polynomial is zero.
Example 1 : Find the degree of each of the polynomials given below:
(i) x
5
– x
4
+ 3 (ii) 2 – y
2
– y
3
+ 2y
8
(iii) 2
Solution : (i) The highest power of the variable is 5. So, the degree of the polynomial
is 5.
(ii) The highest power of the variable is 8. So, the degree of the polynomial is 8.
(iii)The only term here is 2 which can be written as 2x
0
. So the exponent of x is 0.
Therefore, the degree of the polynomial is 0.
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POLYNOMIALS 31
Now observe the polynomials p(x) = 4x + 5, q(y) = 2y, r(t) = t + 2 and
s(u) = 3 – u. Do you see anything common among all of them? The degree of each of
these polynomials is one. A polynomial of degree one is called a linear polynomial.
Some more linear polynomials in one variable are 2x – 1, 2 y + 1, 2 – u. Now, try and
find a linear polynomial in x with 3 terms? You would not be able to find it because a
linear polynomial in x can have at most two terms. So, any linear polynomial in x will
be of the form ax + b, where a and b are constants and a ? 0 (why?). Similarly,
ay + b is a linear polynomial in y.
Now consider the polynomials :
2x
2
+ 5, 5x
2
+ 3x + p, x
2
and x
2
+
2
5
x
Do you agree that they are all of degree two? A polynomial of degree two is called
a quadratic polynomial. Some examples of a quadratic polynomial are 5 – y
2
,
4y + 5y
2
and 6 – y – y
2
. Can you write a quadratic polynomial in one variable with four
different terms? Y ou will find that a quadratic polynomial in one variable will have at
most 3 terms. If you list a few more quadratic polynomials, you will find that any
quadratic polynomial in x is of the form ax
2
+ bx + c, where a ? 0 and a, b, c are
constants. Similarly, quadratic polynomial in y will be of the form ay
2
+ by + c, provided
a ? 0 and a, b, c are constants.
We call a polynomial of degree three a cubic polynomial. Some examples of a
cubic polynomial in x are 4x
3
, 2x
3
+ 1, 5x
3
+ x
2
, 6x
3
– x, 6 – x
3
, 2x
3
+ 4x
2
+ 6x + 7. How
many terms do you think a cubic polynomial in one variable can have? It can have at
most 4 terms. These may be written in the form ax
3
+ bx
2
+ cx + d, where a ? 0 and
a, b, c and d are constants.
Now, that you have seen what a polynomial of degree 1, degree 2, or degree 3
looks like, can you write down a polynomial in one variable of degree n for any natural
number n? A polynomial in one variable x of degree n is an expression of the form
a
n
x
n
+ a
n–1
x
n–1
+ . . . + a
1
x + a
0
where a
0
, a
1
, a
2
, . . ., a
n
are constants and a
n
? 0.
In particular, if a
0
= a
1
= a
2
= a
3
= . . . = a
n
= 0 (all the constants are zero), we get
the zero polynomial, which is denoted by 0. What is the degree of the zero polynomial?
The degree of the zero polynomial is not defined.
So far we have dealt with polynomials in one variable only. We can also have
polynomials in more than one variable. For example, x
2
+ y
2
+ xyz (where variables
are x, y and z) is a polynomial in three variables. Similarly p
2
+ q
10
+ r (where the
variables are p, q and r), u
3
+ v
2
(where the variables are u and v) are polynomials in
three and two variables, respectively. Y ou will be studying such polynomials in detail
later.
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32 MATHEMA TICS
EXERCISE 2.1
1. Which of the following expressions are polynomials in one variable and which are
not? State reasons for your answer.
(i) 4x
2
– 3x + 7 (ii) y
2
+
2
(iii) 3 2 t t + (iv) y +
2
y
(v) x
10
+ y
3
+ t
50
2. Write the coefficients of x
2
in each of the following:
(i) 2 + x
2
+ x (ii) 2 – x
2
+ x
3
(iii)
2
2
x x
p
+ (iv) 2 1 x -
3. Give one example each of a binomial of degree 35, and of a monomial of degree 100.
4. Write the degree of each of the following polynomials:
(i) 5x
3
+ 4x
2
+ 7x (ii) 4 – y
2
(iii) 5t –
7
(iv) 3
5. Classify the following as linear, quadratic and cubic polynomials:
(i) x
2
+ x (ii) x – x
3
(iii) y + y
2
+ 4 (iv) 1 + x
(v) 3t (vi) r
2
(vii) 7x
3
2.3 Zeroes of a Polynomial
Consider the polynomial p(x) = 5x
3
– 2x
2
+ 3x – 2.
If we replace x by 1 everywhere in p(x), we get
p(1) = 5 × (1)
3
– 2 × (1)
2
+ 3 × (1) – 2
= 5 – 2 + 3 –2
= 4
So, we say that the value of p(x) at x = 1 is 4.
Similarly, p(0) = 5(0)
3
– 2(0)
2
+ 3(0) –2
= –2
Can you find p(–1)?
Example 2 : Find the value of each of the following polynomials at the indicated value
of variables:
(i) p(x) = 5x
2
– 3x + 7 at x = 1.
(ii) q(y) = 3y
3
– 4y + 11 at y = 2.
(iii) p(t) = 4t
4
+ 5t
3
– t
2
+ 6 at t = a.
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