Class 8 Exam > Class 8 Notes > Mathematics (Maths) Class 8 > Factorization - Exercise 7.1
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Page 1
Q u e s t i o n : 1
Find the greatest common factor GCF/HCF of the following polynomial:
2x
2
and 12x
2
S o l u t i o n :
The numerical coefficients of the given monomials are 2 and 12. So, the greatest common factor of 2 and 12 is 2.
The common literal appearing in the given monomials is x.
The smallest power of x in the two monomials is 2.
The monomial of the common literals with the smallest powers is x
2
.
Hence, the greatest common factor is 2x
2
.
Q u e s t i o n : 2
Find the greatest common factor GCF/HCF of the following polynomial:
6x
3
y and 18x
2
y
3
S o l u t i o n :
The numerical coefficients of the given monomials are 6 and 18. The greatest common factor of 6 and 18 is 6.
The common literals appearing in the two monomials are x and y.
The smallest power of x in the two monomials is 2.
The smallest power of y in the two monomials is 1.
The monomial of the common literals with the smallest powers is x
2
y.
?Hence, the greatest common factor is 6x
2
y ?.
Q u e s t i o n : 3
Find the greatest common factor GCF/HCF of the following polynomial:
7x, 21x
2
and 14xy
2
S o l u t i o n :
The numerical coefficients of the given monomials are 7, 21 and 14. The greatest common factor of 7, 21 and 14 is 7.
The common literal appearing in the three monomials is x.
The smallest power of x in the three monomials is 1.
The monomial of the common literals with the smallest powers is x.
?Hence, the greatest common factor is 7x.
Q u e s t i o n : 4
Find the greatest common factor GCF/HCF of the following polynomial:
42x
2
yz and 63x
3
y
2
z
3
S o l u t i o n :
The numerical coefficients of the given monomials are 42 and 63. The greatest common factor of 42 and 63 is 21.
The common literals appearing in the two monomials are x, y and z.
The smallest power of x in the two monomials is 2.
The smallest power of y in the two monomials is 1.
The smallest power of z in the two monomials is 1.
The monomial of the common literals with the smallest powers is x
2
yz.
?Hence, the greatest common factor is 21x
2
yz ?.
Q u e s t i o n : 5
Find the greatest common factor GCF/HCF of the following polynomial:
12ax
2
, 6a
2
x
3
and 2a
3
x
5
S o l u t i o n :
The numerical coefficients of the given monomials are 12, 6 and 2. The greatest common factor of 12, 6 and 2 is 2.
The common literals appearing in the three monomials are a and x.
The smallest power of a in the three monomials is 1.
The smallest power of x in the three monomials is 2.
The monomial of common literals with the smallest powers is ax
2
.
?Hence, the greatest common factor is 2ax
2
.
Q u e s t i o n : 6
Find the greatest common factor GCF/HCF of the following polynomial:
9x
2
, 15x
2
y
3
, 6xy
2
and 21x
2
y
2
S o l u t i o n :
The numerical coefficients of the given monomials are 9, 15, 6 and 21. The greatest common factor of 9, 15, 6 and 21 is 3.
The common literal appearing in the three monomials is x.
The smallest power of x in the four monomials is 1.
The monomial of common literals with the smallest powers is x.
?Hence, the greatest common factor is 3x.
Q u e s t i o n : 7
Find the greatest common factor GCF/HCF of the following polynomial:
4a
2
b
3
, -12a
3
b, 18a
4
b
3
S o l u t i o n :
The numerical coefficients of the given monomials are 4, -12 and 18. The greatest common factor of 4, -12 and 18 is 2.
The common literals appearing in the three monomials are a and b.
The smallest power of a in the three monomials is 2.
The smallest power of b in the three monomials is 1.
The monomial of the common literals with the smallest powers is a
2
b.
?Hence, the greatest common factor is 2a
2
b ?.
Q u e s t i o n : 8
Find the greatest common factor GCF/HCF of the following polynomial:
6x
2
y
2
, 9xy
3
, 3x
3
y
2
S o l u t i o n :
The numerical coefficients of the given monomials are 6, 9 and 3. The greatest common factor of 6, 9 and 3 is 3.
Page 2
Q u e s t i o n : 1
Find the greatest common factor GCF/HCF of the following polynomial:
2x
2
and 12x
2
S o l u t i o n :
The numerical coefficients of the given monomials are 2 and 12. So, the greatest common factor of 2 and 12 is 2.
The common literal appearing in the given monomials is x.
The smallest power of x in the two monomials is 2.
The monomial of the common literals with the smallest powers is x
2
.
Hence, the greatest common factor is 2x
2
.
Q u e s t i o n : 2
Find the greatest common factor GCF/HCF of the following polynomial:
6x
3
y and 18x
2
y
3
S o l u t i o n :
The numerical coefficients of the given monomials are 6 and 18. The greatest common factor of 6 and 18 is 6.
The common literals appearing in the two monomials are x and y.
The smallest power of x in the two monomials is 2.
The smallest power of y in the two monomials is 1.
The monomial of the common literals with the smallest powers is x
2
y.
?Hence, the greatest common factor is 6x
2
y ?.
Q u e s t i o n : 3
Find the greatest common factor GCF/HCF of the following polynomial:
7x, 21x
2
and 14xy
2
S o l u t i o n :
The numerical coefficients of the given monomials are 7, 21 and 14. The greatest common factor of 7, 21 and 14 is 7.
The common literal appearing in the three monomials is x.
The smallest power of x in the three monomials is 1.
The monomial of the common literals with the smallest powers is x.
?Hence, the greatest common factor is 7x.
Q u e s t i o n : 4
Find the greatest common factor GCF/HCF of the following polynomial:
42x
2
yz and 63x
3
y
2
z
3
S o l u t i o n :
The numerical coefficients of the given monomials are 42 and 63. The greatest common factor of 42 and 63 is 21.
The common literals appearing in the two monomials are x, y and z.
The smallest power of x in the two monomials is 2.
The smallest power of y in the two monomials is 1.
The smallest power of z in the two monomials is 1.
The monomial of the common literals with the smallest powers is x
2
yz.
?Hence, the greatest common factor is 21x
2
yz ?.
Q u e s t i o n : 5
Find the greatest common factor GCF/HCF of the following polynomial:
12ax
2
, 6a
2
x
3
and 2a
3
x
5
S o l u t i o n :
The numerical coefficients of the given monomials are 12, 6 and 2. The greatest common factor of 12, 6 and 2 is 2.
The common literals appearing in the three monomials are a and x.
The smallest power of a in the three monomials is 1.
The smallest power of x in the three monomials is 2.
The monomial of common literals with the smallest powers is ax
2
.
?Hence, the greatest common factor is 2ax
2
.
Q u e s t i o n : 6
Find the greatest common factor GCF/HCF of the following polynomial:
9x
2
, 15x
2
y
3
, 6xy
2
and 21x
2
y
2
S o l u t i o n :
The numerical coefficients of the given monomials are 9, 15, 6 and 21. The greatest common factor of 9, 15, 6 and 21 is 3.
The common literal appearing in the three monomials is x.
The smallest power of x in the four monomials is 1.
The monomial of common literals with the smallest powers is x.
?Hence, the greatest common factor is 3x.
Q u e s t i o n : 7
Find the greatest common factor GCF/HCF of the following polynomial:
4a
2
b
3
, -12a
3
b, 18a
4
b
3
S o l u t i o n :
The numerical coefficients of the given monomials are 4, -12 and 18. The greatest common factor of 4, -12 and 18 is 2.
The common literals appearing in the three monomials are a and b.
The smallest power of a in the three monomials is 2.
The smallest power of b in the three monomials is 1.
The monomial of the common literals with the smallest powers is a
2
b.
?Hence, the greatest common factor is 2a
2
b ?.
Q u e s t i o n : 8
Find the greatest common factor GCF/HCF of the following polynomial:
6x
2
y
2
, 9xy
3
, 3x
3
y
2
S o l u t i o n :
The numerical coefficients of the given monomials are 6, 9 and 3. The greatest common factor of 6, 9 and 3 is 3.
The common literals appearing in the three monomials are x and y.
The smallest power of x in the three monomials is 1.
The smallest power of y in the three monomials is 2.
The monomial of common literals with the smallest powers is xy
2
.
?Hence, the greatest common factor is 3xy
2
?.
Q u e s t i o n : 9
Find the greatest common factor GCF/HCF of the following polynomial:
a
2
b
3
, a
3
b
2
S o l u t i o n :
The common literals appearing in the three monomials are a and b.
The smallest power of x in the two monomials is 2.
The smallest power of y in the two monomials is 2.
The monomial of common literals with the smallest powers is a
2
b
2
.
?Hence, the greatest common factor is a
2
b
2
.
Q u e s t i o n : 1 0
Find the greatest common factor GCF/HCF of the following polynomial:
36a
2
b
2
c
4
, 54a
5
c
2
, 90a
4
b
2
c
2
S o l u t i o n :
The numerical coefficients of the given monomials are 36, 54 and 90. The greatest common factor of 36, 54 and 90 is 18.
The common literals appearing in the three monomials are a and c.
The smallest power of a in the three monomials is 2.
The smallest power of c in the three monomials is 2.
The monomial of common literals with the smallest powers is a
2
c
2
.
?Hence, the greatest common factor is 18a
2
c
2
.
Q u e s t i o n : 1 1
Find the greatest common factor GCF/HCF of the following polynomial:
x
3
, - yx
2
S o l u t i o n :
The common literal appearing in the two monomials is x.
The smallest power of x in both the monomials is 2.
?Hence, the greatest common factor is x
2
.
Q u e s t i o n : 1 2
Find the greatest common factor GCF/HCF of the following polynomial:
15a
3
, - 45a
2
, - 150a
S o l u t i o n :
The numerical coefficients of the given monomials are 15, -45 and -150. The greatest common factor of 15, -45 and -150 is 15.
The common literal appearing in the three monomials is a.
The smallest power of a in the three monomials is 1.
?Hence, the greatest common factor is 15a.
Q u e s t i o n : 1 3
Find the greatest common factor GCF/HCF of the following polynomial:
2x
3
y
2
, 10x
2
y
3
, 14xy
S o l u t i o n :
The numerical coefficients of the given monomials are 2, 10 and 14. The greatest common factor of 2, 10 and 14 is 2.
The common literals appearing in the three monomials are x and y.
The smallest power of x in the three monomials is 1.
The smallest power of y in the three monomials is 1.
The monomial of common literals with the smallest powers is xy.
?Hence, the greatest common factor is 2xy.
Q u e s t i o n : 1 4
Find the greatest common factor GCF/HCF of the following polynomial:
14x
3
y
5
, 10x
5
y
3
, 2x
2
y
2
S o l u t i o n :
The numerical coefficients of the given monomials are 14, 10 and 2. The greatest common factor of 14, 10 and 2 is 2.
The common literals appearing in the three monomials are x and y.
The smallest power of x in the three monomials is 2.
The smallest power of y in the three monomials is 2.
The monomial of common literals with the smallest powers is x
2
y
2
.
?Hence, the greatest common factor is 2x
2
y
2
.
Q u e s t i o n : 1 5
Find the greatest common factor of the terms in each of the following expression:
5a
4
+ 10a
3
- 15a
2
S o l u t i o n :
Terms are expressions separated by plus or minus signs. Here, the terms are 5a
4
, 10a
3
and 15a
2
.
The numerical coefficients of the given monomials are 5, 10 and 15. The greatest common factor of 5, 10 and 15 is 5.
The common literal appearing in the three monomials is a.
The smallest power of a in the three monomials is 2.
The monomial of common literals with the smallest powers is a
2
.
?Hence, the greatest common factor is 5a
2
.
Q u e s t i o n : 1 6
Find the greatest common factor of the terms in each of the following expression:
2xyz + 3x
2
y + 4y
2
S o l u t i o n :
The expression has three monomials: 2xyz, 3x
2
y and 4y
2
.
Page 3
Q u e s t i o n : 1
Find the greatest common factor GCF/HCF of the following polynomial:
2x
2
and 12x
2
S o l u t i o n :
The numerical coefficients of the given monomials are 2 and 12. So, the greatest common factor of 2 and 12 is 2.
The common literal appearing in the given monomials is x.
The smallest power of x in the two monomials is 2.
The monomial of the common literals with the smallest powers is x
2
.
Hence, the greatest common factor is 2x
2
.
Q u e s t i o n : 2
Find the greatest common factor GCF/HCF of the following polynomial:
6x
3
y and 18x
2
y
3
S o l u t i o n :
The numerical coefficients of the given monomials are 6 and 18. The greatest common factor of 6 and 18 is 6.
The common literals appearing in the two monomials are x and y.
The smallest power of x in the two monomials is 2.
The smallest power of y in the two monomials is 1.
The monomial of the common literals with the smallest powers is x
2
y.
?Hence, the greatest common factor is 6x
2
y ?.
Q u e s t i o n : 3
Find the greatest common factor GCF/HCF of the following polynomial:
7x, 21x
2
and 14xy
2
S o l u t i o n :
The numerical coefficients of the given monomials are 7, 21 and 14. The greatest common factor of 7, 21 and 14 is 7.
The common literal appearing in the three monomials is x.
The smallest power of x in the three monomials is 1.
The monomial of the common literals with the smallest powers is x.
?Hence, the greatest common factor is 7x.
Q u e s t i o n : 4
Find the greatest common factor GCF/HCF of the following polynomial:
42x
2
yz and 63x
3
y
2
z
3
S o l u t i o n :
The numerical coefficients of the given monomials are 42 and 63. The greatest common factor of 42 and 63 is 21.
The common literals appearing in the two monomials are x, y and z.
The smallest power of x in the two monomials is 2.
The smallest power of y in the two monomials is 1.
The smallest power of z in the two monomials is 1.
The monomial of the common literals with the smallest powers is x
2
yz.
?Hence, the greatest common factor is 21x
2
yz ?.
Q u e s t i o n : 5
Find the greatest common factor GCF/HCF of the following polynomial:
12ax
2
, 6a
2
x
3
and 2a
3
x
5
S o l u t i o n :
The numerical coefficients of the given monomials are 12, 6 and 2. The greatest common factor of 12, 6 and 2 is 2.
The common literals appearing in the three monomials are a and x.
The smallest power of a in the three monomials is 1.
The smallest power of x in the three monomials is 2.
The monomial of common literals with the smallest powers is ax
2
.
?Hence, the greatest common factor is 2ax
2
.
Q u e s t i o n : 6
Find the greatest common factor GCF/HCF of the following polynomial:
9x
2
, 15x
2
y
3
, 6xy
2
and 21x
2
y
2
S o l u t i o n :
The numerical coefficients of the given monomials are 9, 15, 6 and 21. The greatest common factor of 9, 15, 6 and 21 is 3.
The common literal appearing in the three monomials is x.
The smallest power of x in the four monomials is 1.
The monomial of common literals with the smallest powers is x.
?Hence, the greatest common factor is 3x.
Q u e s t i o n : 7
Find the greatest common factor GCF/HCF of the following polynomial:
4a
2
b
3
, -12a
3
b, 18a
4
b
3
S o l u t i o n :
The numerical coefficients of the given monomials are 4, -12 and 18. The greatest common factor of 4, -12 and 18 is 2.
The common literals appearing in the three monomials are a and b.
The smallest power of a in the three monomials is 2.
The smallest power of b in the three monomials is 1.
The monomial of the common literals with the smallest powers is a
2
b.
?Hence, the greatest common factor is 2a
2
b ?.
Q u e s t i o n : 8
Find the greatest common factor GCF/HCF of the following polynomial:
6x
2
y
2
, 9xy
3
, 3x
3
y
2
S o l u t i o n :
The numerical coefficients of the given monomials are 6, 9 and 3. The greatest common factor of 6, 9 and 3 is 3.
The common literals appearing in the three monomials are x and y.
The smallest power of x in the three monomials is 1.
The smallest power of y in the three monomials is 2.
The monomial of common literals with the smallest powers is xy
2
.
?Hence, the greatest common factor is 3xy
2
?.
Q u e s t i o n : 9
Find the greatest common factor GCF/HCF of the following polynomial:
a
2
b
3
, a
3
b
2
S o l u t i o n :
The common literals appearing in the three monomials are a and b.
The smallest power of x in the two monomials is 2.
The smallest power of y in the two monomials is 2.
The monomial of common literals with the smallest powers is a
2
b
2
.
?Hence, the greatest common factor is a
2
b
2
.
Q u e s t i o n : 1 0
Find the greatest common factor GCF/HCF of the following polynomial:
36a
2
b
2
c
4
, 54a
5
c
2
, 90a
4
b
2
c
2
S o l u t i o n :
The numerical coefficients of the given monomials are 36, 54 and 90. The greatest common factor of 36, 54 and 90 is 18.
The common literals appearing in the three monomials are a and c.
The smallest power of a in the three monomials is 2.
The smallest power of c in the three monomials is 2.
The monomial of common literals with the smallest powers is a
2
c
2
.
?Hence, the greatest common factor is 18a
2
c
2
.
Q u e s t i o n : 1 1
Find the greatest common factor GCF/HCF of the following polynomial:
x
3
, - yx
2
S o l u t i o n :
The common literal appearing in the two monomials is x.
The smallest power of x in both the monomials is 2.
?Hence, the greatest common factor is x
2
.
Q u e s t i o n : 1 2
Find the greatest common factor GCF/HCF of the following polynomial:
15a
3
, - 45a
2
, - 150a
S o l u t i o n :
The numerical coefficients of the given monomials are 15, -45 and -150. The greatest common factor of 15, -45 and -150 is 15.
The common literal appearing in the three monomials is a.
The smallest power of a in the three monomials is 1.
?Hence, the greatest common factor is 15a.
Q u e s t i o n : 1 3
Find the greatest common factor GCF/HCF of the following polynomial:
2x
3
y
2
, 10x
2
y
3
, 14xy
S o l u t i o n :
The numerical coefficients of the given monomials are 2, 10 and 14. The greatest common factor of 2, 10 and 14 is 2.
The common literals appearing in the three monomials are x and y.
The smallest power of x in the three monomials is 1.
The smallest power of y in the three monomials is 1.
The monomial of common literals with the smallest powers is xy.
?Hence, the greatest common factor is 2xy.
Q u e s t i o n : 1 4
Find the greatest common factor GCF/HCF of the following polynomial:
14x
3
y
5
, 10x
5
y
3
, 2x
2
y
2
S o l u t i o n :
The numerical coefficients of the given monomials are 14, 10 and 2. The greatest common factor of 14, 10 and 2 is 2.
The common literals appearing in the three monomials are x and y.
The smallest power of x in the three monomials is 2.
The smallest power of y in the three monomials is 2.
The monomial of common literals with the smallest powers is x
2
y
2
.
?Hence, the greatest common factor is 2x
2
y
2
.
Q u e s t i o n : 1 5
Find the greatest common factor of the terms in each of the following expression:
5a
4
+ 10a
3
- 15a
2
S o l u t i o n :
Terms are expressions separated by plus or minus signs. Here, the terms are 5a
4
, 10a
3
and 15a
2
.
The numerical coefficients of the given monomials are 5, 10 and 15. The greatest common factor of 5, 10 and 15 is 5.
The common literal appearing in the three monomials is a.
The smallest power of a in the three monomials is 2.
The monomial of common literals with the smallest powers is a
2
.
?Hence, the greatest common factor is 5a
2
.
Q u e s t i o n : 1 6
Find the greatest common factor of the terms in each of the following expression:
2xyz + 3x
2
y + 4y
2
S o l u t i o n :
The expression has three monomials: 2xyz, 3x
2
y and 4y
2
.
The numerical coefficients of the given monomials are 2, 3 and 4. The greatest common factor of 2, 3 and 4 is 1.
The common literal appearing in the three monomials is y.
The smallest power of y in the three monomials is 1.
The monomial of common literals with the smallest powers is y.
?Hence, the greatest common factor is y.
Q u e s t i o n : 1 7
Find the greatest common factor of the terms in each of the following expression:
3a
2
b
2
+ 4b
2
c
2
+ 12a
2
b
2
c
2
S o l u t i o n :
The expression has three monomials: 3a
2
b
2
, 4b
2
c
2
and 12a
2
b
2
c
2
.
The numerical coefficients of the given monomials are 3, 4 and 12. The greatest common factor of 3, 4 and 12 is 1.
The common literal appearing in the three monomials is b.
The smallest power of b in the three monomials is 2.
The monomial of common literals with the smallest powers is b
2
.
?Hence, the greatest common factor is b
2
.
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FAQs on Factorization - Exercise 7.1 - Mathematics (Maths) Class 8
| 1. What is factorization and why is it important in mathematics? |  |
Ans. Factorization is the process of breaking down a number or an expression into its factors, which are the numbers or expressions that multiply together to give the original number or expression. It is important in mathematics because it helps in simplifying complex expressions, solving equations, finding common factors, and understanding the properties of numbers.
| 2. What are prime numbers and how are they related to factorization? | |
Ans. Prime numbers are numbers that are divisible only by 1 and themselves. They have exactly two factors. Factorization is closely related to prime numbers because every positive integer greater than 1 can be expressed as a product of prime numbers, known as its prime factorization. Prime factorization helps in finding the prime factors of a number and understanding its properties.
| 3. How do you factorize a quadratic expression? | |
Ans. To factorize a quadratic expression, we look for two binomials that, when multiplied together, give the original quadratic expression. This can be done by using methods like factoring by grouping, factoring trinomials, or using the quadratic formula. By factorizing a quadratic expression, we can solve quadratic equations, simplify expressions, and understand the behavior of the quadratic function.
| 4. Can you explain the difference between factorization and prime factorization? | |
Ans. Factorization refers to the process of breaking down a number or an expression into its factors, which can be prime or composite numbers. Prime factorization, on the other hand, specifically involves expressing a number as a product of its prime factors. In prime factorization, we decompose a number into its smallest prime factors, whereas in factorization, we decompose it into any combination of prime or composite factors.
| 5. How is factorization used in cryptography and security? | |
Ans. Factorization plays a crucial role in cryptography and security, particularly in the field of public-key encryption. The security of many encryption algorithms relies on the difficulty of factorizing large composite numbers into their prime factors. By using large prime numbers as encryption keys, it becomes computationally hard for an attacker to factorize them and break the encryption. The factorization process is used to ensure the confidentiality and integrity of sensitive information transmitted over networks.
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Full syllabus notes, lecture & questions for Factorization - Exercise 7.1 | Mathematics (Maths) Class 8 - Class 8 | Plus excerises question with solution to help you revise complete syllabus for Mathematics (Maths) Class 8 | Best notes, free PDF download
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Additional Information about Factorization - Exercise 7.1 for Class 8 Preparation
Factorization - Exercise 7.1 Free PDF Download
The Factorization - Exercise 7.1 is an invaluable resource that delves deep into the core of the Class 8 exam.
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Importance of Factorization - Exercise 7.1
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Factorization - Exercise 7.1 Notes
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Factorization - Exercise 7.1 Class 8 Questions
The "Factorization - Exercise 7.1 Class 8 Questions" guide is a valuable resource for all aspiring students preparing for the
Class 8 exam. It focuses on providing a wide range of practice questions to help students gauge
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The content of Factorization - Exercise 7.1 is prepared as per the latest Class 8 syllabus.
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