Class 9 Exam > Class 9 Notes > Mathematics (Maths) Class 9 > RD Sharma Solutions: Factorization of Polynomials- 1
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Page 1
Q u e s t i o n : 1
Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer:
i
3x
2
- 4x + 15
ii
y
2
+ 2v 3
iii
3v x + v 2x
iv
x -
4
x
v
x
12
+ y
3
+ t
50
S o l u t i o n :
i is a polynomial of degree 2.i.e Quadratic polynomial.
ii is a polynomial of degree 2 in y variable. i.e. Quadratic polynomial.
iii
It is not a polynomial because exponent of x is 1/2 which is not a positive integer.
iv
It is not a polynomial because is fractional part.
v
It is a polynomial in three variables x, y and t.
Q u e s t i o n : 2
Write the coefficient of x
2
in each of the following:
i
17 - 2x + 7x
2
ii
9 - 12x + x
3
iii
p
6
x
2
-3x +4
iv
v 3x -7
S o l u t i o n :
i
Coefficient of
ii
Coefficient of
iii
Coefficient of
iv
Coefficient of
Q u e s t i o n : 3
Write the degrees of each of the following polynomials:
i
7x
3
+ 4x
2
- 3x + 12
ii
12 - x + 2x
3
iii
5y -v 2
iv
7
Page 2
Q u e s t i o n : 1
Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer:
i
3x
2
- 4x + 15
ii
y
2
+ 2v 3
iii
3v x + v 2x
iv
x -
4
x
v
x
12
+ y
3
+ t
50
S o l u t i o n :
i is a polynomial of degree 2.i.e Quadratic polynomial.
ii is a polynomial of degree 2 in y variable. i.e. Quadratic polynomial.
iii
It is not a polynomial because exponent of x is 1/2 which is not a positive integer.
iv
It is not a polynomial because is fractional part.
v
It is a polynomial in three variables x, y and t.
Q u e s t i o n : 2
Write the coefficient of x
2
in each of the following:
i
17 - 2x + 7x
2
ii
9 - 12x + x
3
iii
p
6
x
2
-3x +4
iv
v 3x -7
S o l u t i o n :
i
Coefficient of
ii
Coefficient of
iii
Coefficient of
iv
Coefficient of
Q u e s t i o n : 3
Write the degrees of each of the following polynomials:
i
7x
3
+ 4x
2
- 3x + 12
ii
12 - x + 2x
3
iii
5y -v 2
iv
7
v
0
S o l u t i o n :
i 7x
3
+ 4x
2
- 3x + 12
Degree of the polynomial = 3
Because the highest power of x is 3.
ii
Degree of the polynomial = 3. Because the highest power of x is 3.
iii
Degree of the polynomial = 1. Because the highest power of y is 1.
iv 7
Degree of the polynomial = 0. Because there is no variable term in the expression
v 0
Degree of the polynomial is not defined. As there is no variable or constant term
Q u e s t i o n : 4
Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:
i
x + x
2
+ 4
ii
3x - 2
iii
2x + x
2
iv
3y
v
t
2
+ 1
vi
7t
4
+ 4t
3
+ 3t - 2
S o l u t i o n :
i
The degree of the polynomial is 2. It is quadratic in x.
So, it is quadratic polynomial.
ii
The degree of the polynomial is 1. It is a linear polynomial in x.
iii
The degree of the polynomial is 1.
It is linear a polynomial in x.
iv 3y
The degree of the polynomial is 1. It is linear in y.
v
The degree of the polynomial is 2. It is quadratic polynomial in t.
vi
The degree of the polynomial is 4. Therefore, it is bi-quadratic polynomial in t.
Q u e s t i o n : 5
Classify the following polynomials as polynomials in one-variable, two variables etc.:
i
x
2
- xy + 7y
2
ii
x
2
- 2tx + 7t
2
- x + t
iii
t
3
- 3t
2
+ 4t - 5
iv
xy + yz + zx
S o l u t i o n :
i
Here, x and y are two variables.
So, it is polynomial in two variables.
ii
Here, x and t are two variables.
Page 3
Q u e s t i o n : 1
Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer:
i
3x
2
- 4x + 15
ii
y
2
+ 2v 3
iii
3v x + v 2x
iv
x -
4
x
v
x
12
+ y
3
+ t
50
S o l u t i o n :
i is a polynomial of degree 2.i.e Quadratic polynomial.
ii is a polynomial of degree 2 in y variable. i.e. Quadratic polynomial.
iii
It is not a polynomial because exponent of x is 1/2 which is not a positive integer.
iv
It is not a polynomial because is fractional part.
v
It is a polynomial in three variables x, y and t.
Q u e s t i o n : 2
Write the coefficient of x
2
in each of the following:
i
17 - 2x + 7x
2
ii
9 - 12x + x
3
iii
p
6
x
2
-3x +4
iv
v 3x -7
S o l u t i o n :
i
Coefficient of
ii
Coefficient of
iii
Coefficient of
iv
Coefficient of
Q u e s t i o n : 3
Write the degrees of each of the following polynomials:
i
7x
3
+ 4x
2
- 3x + 12
ii
12 - x + 2x
3
iii
5y -v 2
iv
7
v
0
S o l u t i o n :
i 7x
3
+ 4x
2
- 3x + 12
Degree of the polynomial = 3
Because the highest power of x is 3.
ii
Degree of the polynomial = 3. Because the highest power of x is 3.
iii
Degree of the polynomial = 1. Because the highest power of y is 1.
iv 7
Degree of the polynomial = 0. Because there is no variable term in the expression
v 0
Degree of the polynomial is not defined. As there is no variable or constant term
Q u e s t i o n : 4
Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:
i
x + x
2
+ 4
ii
3x - 2
iii
2x + x
2
iv
3y
v
t
2
+ 1
vi
7t
4
+ 4t
3
+ 3t - 2
S o l u t i o n :
i
The degree of the polynomial is 2. It is quadratic in x.
So, it is quadratic polynomial.
ii
The degree of the polynomial is 1. It is a linear polynomial in x.
iii
The degree of the polynomial is 1.
It is linear a polynomial in x.
iv 3y
The degree of the polynomial is 1. It is linear in y.
v
The degree of the polynomial is 2. It is quadratic polynomial in t.
vi
The degree of the polynomial is 4. Therefore, it is bi-quadratic polynomial in t.
Q u e s t i o n : 5
Classify the following polynomials as polynomials in one-variable, two variables etc.:
i
x
2
- xy + 7y
2
ii
x
2
- 2tx + 7t
2
- x + t
iii
t
3
- 3t
2
+ 4t - 5
iv
xy + yz + zx
S o l u t i o n :
i
Here, x and y are two variables.
So, it is polynomial in two variables.
ii
Here, x and t are two variables.
So, it is polynomial in two variables.
iii
Here, only t is variable.
So, it is polynomial in one variable.
iv
Here, x, y and z are three variables
So, it is polynomial in three variables.
Q u e s t i o n : 6
Identify polynomials in the following:
i
f(x) = 4x
3
- x
2
- 3x + 7
ii
g(x) = 2x
3
- 3x
2
+ v x
- 1
iii
p(x) =
2
3
x
2
-
7
4
x +9
iv
q(x) = 2x
2
- 3x +
4
x
+ 2
v
h(x) = x
4
-x
3
2
+x -1
vi
f(x) = 2 +
3
x
+4x
S o l u t i o n :
i
It is cubic in x, so, it is cubic polynomial in x variable.
ii
Here, exponent of x in is not a positive integer, so, it is not a polynomial.
iii
It is a quadratic polynomial.
iv q(x) = 2x
2
- 3x +
4
x
+ 2
Here, exponent of x in is not a positive integer. So it is not a polynomial.
v
Here, exponent of x in x
3/2
is not a positive integer. So, it is not a polynomial.
vi
Here, exponent of x in is not a positive integer, so, it not a polynomial.
Q u e s t i o n : 7
Identify constant, linear, quadratic and cubic polynomials from the following polynomials:
i
f(x) = 0
ii
g(x) = 2x
3
- 7x + 4
iii
h(x) = -3x +
1
2
iv
Page 4
Q u e s t i o n : 1
Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer:
i
3x
2
- 4x + 15
ii
y
2
+ 2v 3
iii
3v x + v 2x
iv
x -
4
x
v
x
12
+ y
3
+ t
50
S o l u t i o n :
i is a polynomial of degree 2.i.e Quadratic polynomial.
ii is a polynomial of degree 2 in y variable. i.e. Quadratic polynomial.
iii
It is not a polynomial because exponent of x is 1/2 which is not a positive integer.
iv
It is not a polynomial because is fractional part.
v
It is a polynomial in three variables x, y and t.
Q u e s t i o n : 2
Write the coefficient of x
2
in each of the following:
i
17 - 2x + 7x
2
ii
9 - 12x + x
3
iii
p
6
x
2
-3x +4
iv
v 3x -7
S o l u t i o n :
i
Coefficient of
ii
Coefficient of
iii
Coefficient of
iv
Coefficient of
Q u e s t i o n : 3
Write the degrees of each of the following polynomials:
i
7x
3
+ 4x
2
- 3x + 12
ii
12 - x + 2x
3
iii
5y -v 2
iv
7
v
0
S o l u t i o n :
i 7x
3
+ 4x
2
- 3x + 12
Degree of the polynomial = 3
Because the highest power of x is 3.
ii
Degree of the polynomial = 3. Because the highest power of x is 3.
iii
Degree of the polynomial = 1. Because the highest power of y is 1.
iv 7
Degree of the polynomial = 0. Because there is no variable term in the expression
v 0
Degree of the polynomial is not defined. As there is no variable or constant term
Q u e s t i o n : 4
Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:
i
x + x
2
+ 4
ii
3x - 2
iii
2x + x
2
iv
3y
v
t
2
+ 1
vi
7t
4
+ 4t
3
+ 3t - 2
S o l u t i o n :
i
The degree of the polynomial is 2. It is quadratic in x.
So, it is quadratic polynomial.
ii
The degree of the polynomial is 1. It is a linear polynomial in x.
iii
The degree of the polynomial is 1.
It is linear a polynomial in x.
iv 3y
The degree of the polynomial is 1. It is linear in y.
v
The degree of the polynomial is 2. It is quadratic polynomial in t.
vi
The degree of the polynomial is 4. Therefore, it is bi-quadratic polynomial in t.
Q u e s t i o n : 5
Classify the following polynomials as polynomials in one-variable, two variables etc.:
i
x
2
- xy + 7y
2
ii
x
2
- 2tx + 7t
2
- x + t
iii
t
3
- 3t
2
+ 4t - 5
iv
xy + yz + zx
S o l u t i o n :
i
Here, x and y are two variables.
So, it is polynomial in two variables.
ii
Here, x and t are two variables.
So, it is polynomial in two variables.
iii
Here, only t is variable.
So, it is polynomial in one variable.
iv
Here, x, y and z are three variables
So, it is polynomial in three variables.
Q u e s t i o n : 6
Identify polynomials in the following:
i
f(x) = 4x
3
- x
2
- 3x + 7
ii
g(x) = 2x
3
- 3x
2
+ v x
- 1
iii
p(x) =
2
3
x
2
-
7
4
x +9
iv
q(x) = 2x
2
- 3x +
4
x
+ 2
v
h(x) = x
4
-x
3
2
+x -1
vi
f(x) = 2 +
3
x
+4x
S o l u t i o n :
i
It is cubic in x, so, it is cubic polynomial in x variable.
ii
Here, exponent of x in is not a positive integer, so, it is not a polynomial.
iii
It is a quadratic polynomial.
iv q(x) = 2x
2
- 3x +
4
x
+ 2
Here, exponent of x in is not a positive integer. So it is not a polynomial.
v
Here, exponent of x in x
3/2
is not a positive integer. So, it is not a polynomial.
vi
Here, exponent of x in is not a positive integer, so, it not a polynomial.
Q u e s t i o n : 7
Identify constant, linear, quadratic and cubic polynomials from the following polynomials:
i
f(x) = 0
ii
g(x) = 2x
3
- 7x + 4
iii
h(x) = -3x +
1
2
iv
p(x) = 2x
2
- x + 4
v
q(x) = 4x + 3
vi
r(x) = 3x
3
+ 4x
2
+ 5x - 7
S o l u t i o n :
i
The given expression is a Constant polynomial as there is no variable term in it.
ii
The given expression is Cubic polynomial as the highest exponent of x is 3.
iii
The given expression is linear polynomial as the highest exponent of x is 1.
iv
The given expression is Quadratic polynomial as the highest exponent of x is 2.
v
The given polynomial is an linear polynomial as the highest exponent of x is 1.
vi
The given polynomial is Cubic polynomial as the highest exponent of x is 3.
Q u e s t i o n : 8
Give one example each of a binomial of degree 35, and of a monomial of degree 100.
S o l u t i o n :
An example of binomial of degree 35 is
f t = 4t
35
-
1
2
. It is a binomial as it has two terms and degree is 35 because highest exponent of t is 35.
An example of monomial of degree 100 is . It is a monomial as it has only one term and degree is 100 because highest exponent of x is 100
( )
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FAQs on Factorization of Polynomials- 1 RD Sharma Solutions - Mathematics (Maths) Class 9
| 1. What is polynomial factorization? |  |
Ans. Polynomial factorization is the process of expressing a polynomial as a product of its factors. It involves finding the factors of the given polynomial and writing it in the form of a product of these factors.
| 2. Why is polynomial factorization important? | |
Ans. Polynomial factorization is important because it helps us understand the behavior and properties of polynomials. It allows us to simplify complex polynomial expressions, solve polynomial equations, and find the roots of the polynomial.
| 3. How do you factorize a polynomial? | |
Ans. To factorize a polynomial, we look for common factors, apply factor theorem, and use various techniques such as grouping, difference of squares, perfect square trinomial, and cubic identities. By identifying the factors and multiplying them, we can express the polynomial as a product of its factors.
| 4. Can all polynomials be factorized? | |
Ans. No, not all polynomials can be factorized. Some polynomials have prime factors that cannot be further simplified. For example, the polynomial x^2 + 1 cannot be factorized further as there are no real factors that can be identified.
| 5. What are the applications of polynomial factorization? | |
Ans. Polynomial factorization has various applications in mathematics and other fields. It is used in solving polynomial equations, simplifying expressions in algebra, finding the roots of a polynomial, and in areas such as physics, engineering, and computer science where polynomial equations are commonly encountered.
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