Chapter 7: Factorization Exercise – 7.2

Question: 1

Factorize:

3x – 9 

Solution:

The greatest common factor of the terms 3x and -9 of the expression 3x – 9 is 3.

Now,

3x = 3x and – 9 = 3(-3)

Hence, the expression 3x – 9 can be factorised as 3(x – 3).

Question: 2

Factorize:

5x – 15x²

Solution:

The greatest common factor of the terms 5x and 15x² of the expression 5x – 15x² is 5x.

Now,

5x = 5x. (1) and -15x² = 5x.(-3x)

Hence, the expression 5x – 15x² can be factorised as 5x (1 – 3x)

Question: 3

Factorize:

20a¹²b² - 15a⁸b⁴ 

Solution:

The greatest common factor of the terms

20a¹²b² and -15a⁸b⁴ of the expression 20a¹²b² – 15a⁸b⁴ is 5a⁸b².

20a¹²b² = 5x4xa⁸xa⁴xb² = 5a⁸xb²x4a⁴ and -15a⁸xb⁴ = 5x(-3)xa⁸xb²xb² = 5a⁸b² x(-3)b²

Hence, the expression 20a¹²b² – 15a⁸b⁴ can be factorised as 5a⁸b²(4a⁴ – 3b²)

Question: 4

Factorize:

72x⁶y⁷ – 96x⁷y⁶

Solution:

The greatest common factor of the terms 72x⁶y⁷ and -96x⁷y⁶ of the expression 72x⁶y⁷- 96x⁷y⁶⁴ is 24x⁶y⁶

Now,

72x⁶y⁷= 24x⁶y⁶. 3y

And, – 96x⁷y⁶⁴ is 24x⁶y⁶. – 4x

Hence, the expression 72x⁶y⁷ -96x⁷y⁶ can be factorised as 24x⁶y⁶. (3y – 4x).

Question: 5

Factorize:

20x³ – 40x² + 80x

Solution:

The greatest common factor of the terms 20x³, -40x² and 80x of the expression 20x³ – 40x² +80x is 20x.

Now, 20x³ = 20x. x²

- 40x² = 20x. -2x And, 80x = 20x. 4

Hence, the expression 20x³ – 40x² + 80x can be factorised as 20x(x² – 2x + 4)

Question: 6

Factorize:

2x³y² – 4x²y³ + 8xy⁴

Solution:

The greatest common factor of the terms 2x³y², -4x²y³ and 8xy⁴ of the expression

2x³y² -4x²y³ + 8xy⁴ is 2xy².

Now,

2x³y² = 2xy². x²

- 4x²y³ = 2xy². (-2xy)

8xy⁴ = 2xy². 4y²

Hence, the expression 2x³y² - 4x²y³ + 8xy⁴ can be factorised as 2xy²(x² – 2xy + 4y²)

Question: 7

Factorize:

10m³n² + 15m⁴n – 20m²n³

Solution:

The greatest common factor of the terms 10³n², 15m⁴n and -20m²n³ of the expression

10m³n² + 15m⁴n – 20m²n³ is 5m²n.

Now,

10m³n² = 5m²n. 2mn

15m⁴n = 5m²n. 3m²

-20m²n³ = 5m²n. - 4n²

Hence, 10m³n² + 15m²n – 20m²n³ can be factorised as 5m²n(2mn + 3m² – 4n²)

Question: 8

Factorize:

2a⁴b⁴ – 3a³b⁵ + 4a²b⁵

Solution:

The greatest common factor of the terms 2a⁴b⁴, -3a³b⁵ and 4a²b⁵ of the expression

2a⁴b⁴ – 3a³b⁵ + 4a²b⁵ is a²b⁵.

Now,

2a⁴b⁴ = a²b⁵. 2a²

-3a³b⁵ = a²b⁴. (-3ab)

4a²b⁵ = a²b⁴. 4b

Hence, 2a⁴b⁴- 3a³b⁵ + 4a²b⁵ can be factorised as a²b⁴(2a² – 3ab + 4b)

Question: 9

Factorize:

28a² + 14a²b² – 21a⁴

Solution:

The greatest common factor of the terms28a², 14a²b² and 21a⁴ of the expression

28a² + 14a²b² – 21a⁴ is 7a².

Also, we can write 28a² = 7a². 4, 14a²b² = 7a². 2b² and 21a⁴ = 7a². 3a².

Therefore, 28a² + 14a²b² – 21a⁴ = 7a². 4 + 7a². 2b² – 7a². 3a²

= 7a² (4 + 2b² – 3a²)

Question: 10

Factorize:

 a⁴b – 3a²b² – 6ab³

Solution:

The greatest common factor of the terms a⁴b, 3a²b² and 6ab³ of the expression

a⁴b – 3a²b² – 6ab³ is ab.

Also, we can write a⁴b = ab. a³, 3a²b² = ab. 3ab and 6ab³ = ab. 6b².

Therefore, a⁴b – 3a²b² – 6ab³ = ab. a³ – ab. 3ab – ab. 6b².

= ab (a³ – 3ab – 6b²)

Question: 11

Factorize:

 2L²mn – 3Lm²n + 4Lmn²

Solution:

The greatest common factor of the terms 2L²mn, 3Lm²n and 4Lmn² of the expression

2L²mn – 3Lm²n + 4Lmn² is Lmn.

Also, we can write 2L²mn = Lmn. 2L, 3Lm²n = Lmn. 3m and 4Lmn² = Lmn. 4n

Therefore, 2L²mn - 3Lm²n + 4Lmn² = (Lmn. 2L) – (Lmn. 3m) + (Lmn. 4n)

= Lmn(2L – 3m + 4n)

Question: 12

Factorize:

x⁴y² – x²y⁴ – x⁴y⁴

Solution:

The greatest common factor of the terms x⁴y², x²y⁴ and x⁴y⁴ of the expressinon

x⁴y² – x²y⁴ – x⁴y⁴ is x²y²

Also, we can write  x⁴y²  =  (x²y² . x²) ,  x²y⁴ = (x²y² . y²)  and  x⁴y⁴ =  (x²y² . x²y²)

Therefore, x⁴y² – x²y⁴– x⁴y⁴ = (x²y². x²) – (x²y². y²) – (x²y². x²y²)

= x²y² (x² – y² – x²y²)

Question: 13

Factorize:

9x²y + 3axy

Solution:

The greatest common factor of the terms 9x²y and 3axy of the expression 9x²y + 3axy is 3xy.

Also, we can write 9x²y = 3xy. 3x and 3axy = 3xy.a 

Therefore, 9x²y + 3axy = (3xy. 3x) + (3xy. a)

= 3xy (3x + a)

Question: 14

Factorize:

16m – 4m²

Solution: 

The greatest common factor of the terms 16m and 4m² of the expression 16m – 4m² is 4m.

Also, we can write 16m = 4m. 4 and 4m² = 4m. m

Therefore, 16m – 4m² = (4m. 4) – (4m. m)

= 4m(4 – m)

Question: 15

Factorize:

-4a² + 4ab – 4ca

Solution:

The greatest common factor of the terms - 4a², 4ab and -4ca of the expression

-4a² + 4ab – 4ca is -4a.

Also, we can write -4a² = (-4a. a), 4ab = -4a. (-b), and 4ca = (- 4a. c)

Therefore, -4a² + 4ab – 4ca = (- 4a. a) + (- 4a. (-b)) – (4a. c)

= - 4a (a – b + c)

Question: 16

Factorize:

x²yz + xy²z + xyz²  

Solution:

The greatest common factor of the terms x²yz, xy²z and xyz² of the expression

x²yz + xy²z + xyz² is xyz.

Also, we can write x²yz = (xyz. x), (xy²z = xyz. y), xyz² = (xyz. z)

Therefore, x²yz + xy²z + xyz² = (xyz. x) + (xyz. y) + (xyz. z)

= xyz(x + y + z)

Question: 17

Factorize:

 ax²y + bxy² + cxyz

Solution:

The greatest common factor of the terms ax²y, bxy² and cxyz of the expression

ax²y + bxy² + cxyz is xy.

Also, we can write ax²y = (xy. ax), bxy² = (xy. by), cxyz = (xy. cz)

Therefore, ax²y + bxy² + cxyz = (xy. ax) + (xy. by) + (xy. cz)

= xy (ax + by + cz)