RD Sharma solution class 8 chapter 7 Factorization Exercise 7.5

Exercise 7.5

Page-7.17

Question 1:

Factorize each of the following expression:
16x2 − 25y2

Answer 1:

16x2-25y2=(4x)2-(5y)2=(4x-5y)(4x+5y)

Question 2:

Factorize each of the following expression:
27x2 − 12y2

Answer 2:

27x2-12y2=3(9x2-4y2)=3[(3x)2-(2y)2]=3(3x-2y)(3x+2y)

Question 3:

Factorize each of the following expression:
144a2 − 289b2

Answer 3:

144a2-289b2=(12a)2-(17b)2=(12a-17b)(12a+17b)

Question 4:

Factorize each of the following expression:
12m2 − 27

Answer 4:

12m2-27=3(4m2-9)=3[(2m)2-32]=3(2m-3)(2m+3)

Question 5:

Factorize each of the following expression:
125x2 − 45y2

Answer 5:

125x2-45y2=5(25x2-9y2)=5[(5x)2-(3y)2]=5(5x-3y)(5x+3y)

Question 6:

Factorize each of the following expression:
144a2 − 169b2

Answer 6:

144a2-169b2=(12a)2-(13b)2=(12a-13b)(12a+13b)

Question 7:

Factorize each of the following expression:
(2a − b)2 − 16c2

Answer 7:

(2a-b)2-16c2=(2a-b)2-(4c)2=[(2a-b)-4c][(2a-b)+4c]=(2a-b-4c)(2a-b+4c)

Question 8:

Factorize each of the following expression:
(x + 2y)2 − 4(2x − y)2

Answer 8:

(x+2y)2-4(2x-y)2=(x+2y)2-[2(2x-y)]2                                   =[(x+2y)-2(2x-y)][(x+2y)+2(2x-y)]                                   =(x+2y-4x+2y)(x+2y+4x-2y)                                   =5x(4y-3x)

Question 9:

Factorize each of the following expression:
3a5 − 48a3

Answer 9:

3a5-48a3=3a3(a2-16)=3a3(a2-42)=3a3(a-4)(a+4)

Question 10:

Factorize each of the following expression:
a4 − 16b4

Answer 10:

a4-16b4=a4-24b4=(a2)2-(22b2)2                                     =(a2-22b2)(a2+22b2)                                     =[a2-(2b)2](a2+4b2)                                     =(a-2b)(a+2b)(a2+4b2)

Question 11:

Factorize each of the following expression:
x8 − 1

Answer 11:

x8-1=(x4)2-12=(x4-1)(x4+1)=[(x2)2-12](x4+1)=(x2-1)(x2+1)(x4+1)=(x2-12)(x2+1)(x4+1)=(x-1)(x+1)(x2+1)(x4+1)

Question 12:

Factorize each of the following expression:
64 − (a + 1)2

Answer 12:

64-(a+1)2=(8)2-(a+1)2=[8-(a+1)][8+(a+1)]=(8-a-1)(8+a+1)=(7-a)(9+a)

Question 13:

Factorize each of the following expression:
36l2 − (m + n)2

Answer 13:

36l2-(m+n)2=(6l)2-(m+n)2=[6l-(m+n)][6l+(m+n)]=(6l-m-n)(6l+m+n)

Question 14:

Factorize each of the following expression:
25x4y4 − 1

Answer 14:

25x4y4-1=(5x2y2)2-1=(5x2y2-1)(5x2y2+1)

Question 15:

Factorize each of the following expression:
a4-1b4

Answer 15:

   a1/b4
(a2)1/(b2)2
a2- 1/b2a2 1/b2
1/ba 1/ba2 1/b2

Question 16:

Factorize each of the following expression:
x3 − 144x

Answer 16:

x3-144x=x(x2-144)=x(x2-122)=x(x-12)(x+12)

Question 17:

Factorize each of the following expression:
(x - 4y)2 − 625

Answer 17:

(x-4y)2-625=(x-4y)2-252=[(x-4y)-25][(x-4y)+25]=(x-4y-25)(x-4y+25)

Question 18:

Factorize each of the following expression:
9(a − b)2 − 100(x − y)2

Answer 18:

9(a-b)2-100(x-y)2=[3(a-b)]2-[10(x-y)]2=[3(a-b)-10(x-y)][3(a-b)+10(x-y)]=(3a-3b-10x+10y)(3a-3b+10x-10y)

Question 19:

Factorize each of the following expression:
(3 + 2a)2 − 25a2

Answer 19:

(3+2a)2-25a2=(3+2a)2-(5a)2=[(3+2a)-5a][(3+2a)+5a]=(3+2a-5a)(3+2a+5a)=(3-3a)(3+7a)=3(1-a)(3+7a)

Question 20:

Factorize each of the following expression:
(x + y)2 − (a − b)2

Answer 20:

(x+y)2-(a-b)2=[(x+y)-(a-b)][(x+y)+(a-b)]=(x+y-a+b)(x+y+a-b)

Question 21:

Factorize each of the following expression:
116x2y2-449y2z2

Answer 21:

116x2y2-449y2z2=y2116x2-449z2=y214x2-27z2=y214x-27z14x+27z=y2x4-27zx4+27z

Question 22:

Factorize each of the following expression:
75a3b2 - 108ab4

Answer 22:

75a3b2-108ab4=3ab2(25a2-36b2)=3ab2[(5a)2-(6b)2]=3ab2(5a-6b)(5a+6b)

Question 23:

Factorize each of the following expression:
x5 − 16x3

Answer 23:

x5-16x3=x3(x2-16)=x3(x2-42)=x3(x-4)(x+4)

Question 24:

Factorize each of the following expression:
50x2-2x281

Answer 24:

 50x2-2x281=225x2-x281=25x2-x92=25x-x95x+x9

Question 25:

Factorize each of the following expression:
256x5 − 81x

Answer 25:

256x5-81x=x(256x4-81)=x[(16x2)2-92]=x(16x2+9)(16x2-9)=x(16x2+9)[(4x)2-32]=x(16x2+9)(4x+3)(4x-3)

Question 26:

Factorize each of the following expression:
a4 − (2b + c)4

Answer 26:

a4-(2b+c)4=(a2)2-[(2b+c)2]2=[a2+(2b+c)2][a2-(2b+c)2]=[a2+(2b+c)2]{[a+(2b+c)][a-(2b+c)]}=[a2+(2b+c)2](a+2b+c)(a-2b-c)

Question 27:

Factorize each of the following expression:
(3x + 4y)4x4

Answer 27:

(3x+4y)4-x4=[(3x+4y)2]2-(x2)2=[(3x+4y)2+x2][(3x+4y)2-x2]=[(3x+4y)2+x2][(3x+4y)+x][(3x+4y)-x]={(3x+4y)2+x2}(3x+4y+x)(3x+4y-x)=3x+4y2+x2(4x+4y)(2x+4y)=3x+4y2+x24(x+y)2(x+2y)=83x+4y2+x2(x+y)(x+2y)

Question 28:

Factorize each of the following expression:
p2q2p4q4

Answer 28:

p2q2-p4q4=p2q2(1-p2q2)=p2q2[1-(pq)2]=p2q2(1-pq)(1+pq)

Question 29:

Factorize each of the following expression:
3x3y − 243xy3

Answer 29:

 3x3y-243xy3=3xy(x2-81y2)=3xy[x2-(9y)2]=3xy(x-9y)(x+9y)

Question 30:

Factorize each of the following expression:
a4b4 − 16c4

Answer 30:

a4b4-16c4=[(a2b2)2-(4c2)2]=(a2b2+4c2)(a2b2-4c2)=(a2b2+4c2)[(ab)2-(2c)2]=(a2b2+4c2)(ab+2c)(ab-2c)

Question 31:

Factorize each of the following expression:
x4 − 625

Answer 31:

x4-625=(x2)2-252=(x2+25)(x2-25)=(x2+25)(x2-52)=(x2+25)(x+5)(x-5)

Question 32:

Factorize each of the following expression:
x4 − 1

Answer 32:

 x4-1=(x2)2-1=(x2+1)(x2-1)=(x2+1)(x+1)(x-1)

Question 33:

Factorize each of the following expression:
49(a − b)2 − 25(a + b)2

Answer 33:

49(a-b)2-25(a+b)2=[7(a-b)]2-[5(a+b)]2=[7(a-b)-5(a+b)][7(a-b)+5(a+b)]=(7a-7b-5a-5b)(7a-7b+5a+5b)=(2a-12b)(12a-2b)=2(a-6b)2(6a-b)=4(a-6b)(6a-b)

Question 34:

Factorize each of the following expression:
x − yx2 + y2

Answer 34:

x-y-x2+y2=(x-y)+(y2-x2)               [Regrouping the terms]=(x-y)+(y+x)(y-x)=(x-y)-(y+x)(x-y)        [(y-x)=-(x-y)]=(x-y)[1-(y+x)]=(x-y)(1-x-y)

Question 35:

Factorize each of the following expression:
16(2x − 1)2 − 25y2

Answer 35:

16(2x-1)2-25y2=[4(2x-1)]2-(5y)2=[4(2x-1)-5y][4(2x-1)+5y]=(8x-4-5y)(8x-4+5y)=(8x-5y-4)(8x+5y-4)

Question 36:

Factorize each of the following expression:
4(xy + 1)2 − 9(x − 1)2

Answer 36:

4(xy+1)2-9(x-1)2=[2(xy+1)]2-[3(x-1)]2=[2(xy+1)-3(x-1)][2(xy+1)+3(x-1)]=(2xy+2-3x+3)(2xy+2+3x-3)=(2xy-3x+5)(2xy+3x-1)

Question 37:

Factorize each of the following expression:
(2x + 1)2 − 9x4

Answer 37:

(2x+1)2-9x4=(2x+1)2-(3x2)2=[(2x+1)-3x2][(2x+1)+3x2]=(-3x2+2x+1)(3x2+2x+1)We can factorise the quadratic expressions in the curved brackets as:(-3x2+3x-x+1)(3x2+2x+1)=3x(-x+1)+1(-x+1)(3x2+2x+1)=(-x+1)(3x+1)(3x2+2x+1)=-(x-1)(3x+1)(3x2+2x+1)

Question 38:

Factorize each of the following expression:
x4 − (2y − 3z)2

Answer 38:

 x4-(2y-3z)2=(x2)2-(2y-3z)2=[x2-(2y-3z)][x2+(2y-3z)]=(x2-2y+3z)(x2+2y-3z)

Question 39:

Factorize each of the following expression:
a2b2 + a − b

Answer 39:

a2-b2+a-b=(a2-b2)+(a-b)            [Grouping the terms]                         =(a+b)(a-b)+(a-b)                         =(a-b)(a+b+1)             [Taking out the common factor (a-b)]

Question 40:

Factorize each of the following expression:
16a4b4

Answer 40:

16a4-b4=(4a2)2-(b2)2=(4a2+b2)(4a2-b2)=(4a2+b2)[(2a)2-b2]=(4a2+b2)(2a+b)(2a-b)

Question 41:

Factorize each of the following expression:
a4 − 16(b − c)4

Answer 41:

a4-16(b-c)4=(a2)2-[4(b-c)2]2=[a2+4(b-c)2][a2-4(b-c)2]=[a2+4(b-c)2]{a2-[2(b-c)]2}=[a2+4(b-c)2][a+2(b-c)][a-2(b-c)]=[a2+4(b-c)2](a+2b-2c)(a-2b+2c)

Question 42:

Factorize each of the following expression:
2a5 − 32a

Answer 42:

2a5-32a=2a(a4-16)=2a[(a2)2-42]=2a(a2+4)(a2-4)=2a(a2+4)(a2-22)=2a(a2+4)(a+2)(a-2)=2a(a-2)(a+2)(a2+4)

Question 43:

Factorize each of the following expression:
a4b4 − 81c4

Answer 43:

a4b4-81c4=(a2b2)2-(9c2)2=(a2b2+9c2)(a2b2-9c2)=(a2b2+9c2)[(ab)2-(3c)2]=(a2b2+9c2)(ab+3c)(ab-3c)

Question 44:

Factorize each of the following expression:
xy9yx9

Answer 44:

xy9-yx9=xy(y8-x8)=xy[(y4)2-(x4)2]=xy(y4+x4)(y4-x4)=xy(y4+x4)[(y2)2-(x2)2]=xy(y4+x4)(y2+x2)(y2-x2)=xy(y4+x4)(y2+x2)(y+x)(y-x)

Question 45:

Factorize each of the following expression:
x3x

Answer 45:

x3-x=x(x2-1)=x(x-1)(x+1)

Question 46:

Factorize each of the following expression:
18a2x2 − 32

Answer 46:

 18a2x2-32=2(9a2x2-16)=2[(3ax)2-42]=2(3ax-4)(3ax+4)

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