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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 - 15a8b

Solution:

The greatest common factor of the terms

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

20a12b2 = 5x4xa8xa4xb2 = 5a8xb2x4a4 and -15a8xb= 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

2x3y-4x2y3 + 8xy4 is 2xy2.

Now,

2x3y2 = 2xy2. x2

- 4x2y3 = 2xy2. (-2xy)

8xy4 = 2xy2. 4y2

Hence, the expression 2x3y- 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)


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