Class 8 RD Sharma Solutions - Chapter 7 Factorization - Exercise 7.1

Question 1. Find (GCF/HCF) of 2x²  and 12x²

Solution: 

First Prime Factorize each monomial, then find common among them.

Here,

2x² = 2 * x * x

12x² = 2* 2 * 3 * x * x

GCF(2x², 12x²) = 2 * x * x

                          = 2x²

Question 2. Find (GCF/HCF) of 6x³y and 18x²y³

Solution:  

First Prime Factorize each monomial, then find common among them.

Here,

6x³y = 2 * 3 * x * x * x * y

18x²y³ = 2 * 2 * 3 * x * x * y * y * y

GCF(6x³y, 18x²y³) = 2 * 3 * x * x * y

                               = 6x²y

Question 3. Find (GCF/HCF) of 7x, 21x² and 14xy²

Solution:  

First Prime Factorize each monomial, then find common among them.

Here,

7x = 7 * x

21x² = 7 * 3 * x * x

14xy² = 7 * 2 * x * y * y

GCF(7x, 21x², 14xy²) = 7*x

                                    = 7x

Question 4. Find (GCF/HCF) of 42x²yz and 63x³y²z³

Solution:  

First Prime Factorize each monomial, then find common among them.

Here,

42x²yz = 2 * 3 * 7 * x * x * y * z

63x³y²z³ = 3 * 3 * 7 * x * x * x * y * y * z * z * z

GCF(42x²yz, 63x³y²z³) = 3 * 7 * x * x * y * z

                                       = 21x²yz

Question 5. Find (GCF/HCF) of 12ax², 6a²x³ and 2a³x⁵

Solution:  

First Prime Factorize each monomial, then find common among them.

Here,

12ax² = 2 * 2 * 3 * a * x * x

6a²x³ = 2 * 3 * a * a * x * x * x

2a³x⁵ = 2 * a * a * a * x * x * x * x * x

GCF(12ax², 6a²x³,2a³x⁵) = 2 * a * x * x

                                         = 2ax²

Question 6. Find (GCF/HCF) of 9x², 15x²y³, 6xy² and 21x²y²

Solution:  

First Prime Factorize each monomial, then find common among them.

Here,

9x² = 3 * 3 * x * x

15x²y³ = 3 * 5 * x * x * y * y * y

6xy² = 2 * 3 * x * y * y

21x²y² = 3 * 7 * x * x * y * y

GCF(9x², 15x2y³, 6xy² and 21x²y²) = 3 * x

                                                           = 3x

Question 7. Find (GCF/HCF) of 4a²b³, -12a³b, 18a⁴b³

Solution:  

First Prime Factorize each monomial, then find common among them.

Here,

4a²b³ = 2 * 2 * a * a * b * b * b

-12a³b = -1 * 2 * a * a * a * b

18a⁴b³ = 2 * 3 * 3 * a * a * a * a * b * b * b

GCF(4a²b³, -12a³b, 18a⁴b³) = 2 * a * a * b

                                              = 2a²b                                                    

Question 8. Find (GCF/HCF) of 6x²y², 9xy³, 3x³y²

Solution:  

First Prime Factorize each monomial, then find common among them.

Here,

6x²y² = 2 * 3 * x * x * y * y

9xy³ = 3 * x * y * y * y

3x³y² = 3 * x * x * x * y * y

GCF(6x²y², 9xy³, 3x³y²) = 3 * x * y * y

                                        = 3xy²                                           

Question 9. Find (GCF/HCF) of a²b³,  a³b²

Solution:  

First Prime Factorize each monomial, then find common among them.

Here,

a²b³ = a * a * b * b * b

a³b² = a * a * a * b * b

GCF(a²b³, a³b²) = a * a * b * b

                           = a²b²

Question 10. Find (GCF/HCF) of 36a²b²c⁴,  54a⁵c², 90a⁴b²c²

Solution:  

First Prime Factorize each monomial, then find common among them.

Here,

36a²b²c⁴ = 2 * 2 * 3 * 3 * a * a * b * b * c * c * c * c

54a⁵c² = 2 * 3 * 3 * 3 * a * a * a * a * a * c * c

90a⁴b²c² = 2 * 3 * 3 * 5 * a * a * a * a * b * b * c * c

GCF(36a²b²c⁴, 54a⁵c², 90a⁴b²c²) = 2 * 3 * 3 * a * a * c * c

                                                      = 18a²c²

Question 11. Find (GCF/HCF) of x³, -yx²

Solution:  

First Prime Factorize each monomial, then find common among them.

Here,

x³ = x * x * x

-yx² = -1 * y * x * x

GCF(x³, -yx²) = x * x

                       = x²

Question 12. Find (GCF/HCF) of 15a³, -45a², -150a

Solution:  

First Prime Factorize each monomial, then find common among them.

Here,

15a³ = 3 * 5 * a * a * a

-45a² = -1 * 3 * 3 * 5 * a *a

-150a = -1 * 2 * 3 * 5 * 5 * a

GCF(15a³, -45a², -150a) = 3 * 5 * a

                                        = 15a

Question 13. Find (GCF/HCF) of 2x³y², 10x²y³, 14xy

Solution:  

First Prime Factorize each monomial, then find common among them.

Here,

2x³y² = 2 * x * x * x * y * y

10x²y³ = 2 * 5 * x * x * y * y * y

14xy = 2 * 7 * x * y

GCF(2x³y², 10x²y³, 14xy) = 2 * x * y

                                           = 2xy

Question 14. Find (GCF/HCF) of 14x³y⁵, 10x⁵y³, 2x²y²

Solution:  

First Prime Factorize each monomial, then find common among them.

Here,

14x³y⁵ = 2 * 7 * x * x * x * y * y * y * y * y

10x⁵y³ = 2 * 5 * x * x * x * x * x * y * y * y

2x²y² = 2 * x * x * y * y

GCF(14x³y⁵, 10x⁵y³, 2x²y²) = 2 * x * x * y * y

                                              = 2x²y²

Find the greatest common factor(GCF) of the terms in the following expressions. (Q15 - Q17)

Question 15. 5a⁴ + 10a³ - 15a²

Solution:

Here, 5a⁴ = 5 * a * a * a * a

         10a³ = 2 * 5 * a * a * a

         -15a² = -1 * 3 * 5 * a * a

5a⁴ + 10a³ - 15a² = (5 * a * a * a * a) + (2 * 5 * a * a * a) + (-1 * 3 * 5 * a * a)

                            = 5a²(a² + 2a - 3)

GCF(5a⁴ + 10a³ - 15a²) = 5a²

Question 16. 2xyz + 3x²y + 4y²

Solution:

Here, 2xyz = 2 * x * y * z

          3x²y = 3 * x * x * y

           4y² = 2 * 2 * y * y

2xyz + 3x²y + 4y² = (2 * x * y * z) + (3 * x * x * y) + (4 * y * y)

                             = y(2x + 3x² + 4y)

GCF(2xyz + 3x²y + 4y²) = y

Question 17. 3a²b² + 4b²c² + 12a²b²c²

Solution:

Here, 3a²b² = 3 * a * a * b * b

          4b²c² = 2 * 2 * b * b * c * c

          12a²b²c² = 2 * 2 * 3 * a * a * b * b * c * c

3a²b² + 4b²c² + 12a²b²c² = (3 * a * a * b * b) + (2 * 2 * b * b * c * c) + (2 * 2 * 3 * a * a * b * b * c * c)

                                         = b(3a²b + 4bc² + 12a²bc²)

GCF(3a²b² + 4b²c² + 12a²b²c²) = b  

Summary

Exercise 7.1 in the RD Sharma Class 8 textbook is designed to help students develop a strong foundation in factorizing linear expressions. The practice questions cover a wide range of scenarios, including expressions with a single variable, expressions with multiple variables, and expressions involving common factors. By working through these questions, students will learn to identify the appropriate factorization techniques, apply them correctly, and simplify the given expressions. Mastering the skills covered in this exercise will enable students to tackle more complex algebraic problems and lay the groundwork for further exploration of factorization concepts.