MATHEMATICS IN EVERYDAY LIFE–8

MATHEMATICS IN EVERYDAY LIFE?8

Chapter 7 : Factorisation

ANSWER KEYS

EXERCISE 7.1

1. (i) 6x + 9 we can write, 6x = 2 ? 3 ? x 9 =3 ? 3 6x + 9 = 2 ? 3 ? x + 3 ? 3 = 3 ? (2x + 3)

6x + 9 = 3(2x + 3) (ii) 21x + 14

we can write, 21x = 3 ? 7 ? x 14 = 2 ? 7

21x + 14 = 3 ? 7 ? x + 2 ? 7 = 7(3x + 2)

21x + 14 = 7(3x + 2) (iii) 12x2 ? 9x

we can write, 12x2 = 2 ? 2 ? 3 ? x ? x 9x = 3 ? 3 ? x

12x2 ? 9x = 2 ? 2 ? 3 ? x ? x ? 3 ? 3 ? x = 3 ? x(2 ? 2 ? x ? 3)

12x2 ? 9x = 3x(4x ? 3) 2. (i) 8x2 ? 24xy

we can write, 8x2 = 2 ? 2 ? 2 ? x ? x

24xy = 2 ? 2 ? 2 ? 3 ? x ? y 8x2 ? 24xy = 2 ? 2 ? 2 ? x(x ? 3y)

= 8x(x ? 3y) 8x2 ? 24xy = 8x(x ? 3y) (ii) 10xy2 ? 15x2y

we can write, 10xy2 = 2 ? 5 ? x ? y ? y 15x2y = 3 ? 5 ? x ? x ? y

10xy2 ? 15x2y = 2 ? 5 ? x ? y ? y ? 3 ? 5 ? x ? x ? y = 5xy (2y ? 3x)

10xy2 ? 15x2y = 5xy(2y ? 3x) (iii) 27x3 ? 18x2

we can write,

Mathematics In Everyday Life-8

27x3 = 3 ? 3 ? 3 ? x ? x ? x 18x2 = 2 ? 3 ? 3 ? x ? x 27x3 ? 18x2 =3 ? 3 ? 3 ? x ? x ? x ? 2 ? 3 ? 3 ? x ? x

= 3 ? 3 ? x ? x (3x ? 2) = 9x2(3x ? 2) 27x3 ? 18x2 = 9x2(3x ? 2) 3. (i) 6xy ? 9yz we can write, 6xy = 2 ? 3 ? x ? y 9yz = 3 ? 3 ? y ? z 6xy ? 9yz = 2 ? 3 ? x ? y ? 3 ? 3 ? y ? z = 3y(2x ? 3z) 6xy ? 9yz = 3y(2x ? 3z) (ii) 6x + 18 we can write, 6x = 2 ? 3 ? x 18 = 2 ? 3 ? 3 6x + 18 = 2 ? 3 ? x + 2 ? 3 ? 3 = 2 ? 3(x + 3) 6x + 18 = 6(x + 3) (iii) 10ax ? 15ay we can write, 10ax = 2 ? 5 ? a ? x 15ay = 3 ? 5 ? a ? y 10ax ? 15ay = 2 ? 5 ? a ? x ? 3 ? 5 ? a ? y

= 5 ? a(2x ? 3y) 10ax ? 15ay = 5a(2x ? 3y) 4. (i) x2 + 3x we can write,

x2 = x ? x 3x = 3 ? x x2 + 3x = x ? x + 3 ? x

= x(x + 3) x2 + 3x = x(x + 3) (ii) 21m2 ? 14m

we can write, 21m2 = 3 ? 7 ? m ? m 14m = 2 ? 7 ? m

21m2 ? 14m = 3 ? 7 ? m ? m ? 2 ? 7 ? m = 7 ? m(3m ? 2)

21m2 ? 14m = 7m(3m ? 2)

1

(iii) 9y2 + 6y

we can write, 9y2 = 3 ? 3 ? y ? y 6y = 2 ? 3 ? y

9y2 + 6y = 3 ? 3 ? y ? y + 2 ? 3 ? y = 3 ? y ? (3y + 2)

9y2 + 6y = 3y(3y + 2) (iv) 16x2 ? 24

we can write, 16x2 = 2 ? 2 ? 2 ? 2 ? x ? x 24 = 2 ? 2 ? 2 ? 3

16x2 ? 24 = 2 ? 2 ? 2 ? 2 ? x ? x ? 2 ? 2 ? 2 ? 3 = 2 ? 2 ? 2 ? (2x2 ? 3) = 8(2x2 ? 3)

16x2 ? 24 = 8(2x2 ? 3) 5. (i) ?3x2 + 3xy ? 4xz

we can write, 3x2 = 3 ? x ? x 3xy = 3 ? x ? y 4xz = 2 ? 2 ? x ? z

? 3x2 + 3xy ? 4xz = ?3 ? x ? x + 3 ? x ? y ? 2 ? 2 ? x ? z

= ?x(3x ? 3y + 4z) ?3x2 + 3xy ? 4xz = ?x(3x ? 3y + 4z) (ii) x2yz + xy2z + xyz2 we can write,

x2yz = x ? x ? y ? z xy2z = x ? y ? y ? z xyz2 = x ? y ? z ? z x2yz + xy2z + xyz2 =(x ? x ? y ? z + x ? y ? y ? z

+ x ? y ? z ? z) = xyz(x + y + z) x2yz + xy2z + xyz2 = xyz(x + y + z) (iii) x2y ? xy2 + x2y2 we can write, x2y = x ? x ? y xy2 = x ? y ? y x2y2 = x ? x ? y ? y x2y ? xy2 + x2y2 = x ? x ? y ? x ? y ? y

+ x ? x ? y ? y = xy(x ? y + xy) x2y ? xy2 + x2y2 = xy(x ? y + xy) 6. 14a2 ? 28a2b2 + 21a4

we can write, 14a2 = 2 ? 7 ? a ? a

28a2b2 = 2 ? 2 ? 7 ? a ? a ? b ? b 21a4 = 3 ? 7 ? a ? a ? a ? a

14a2 ?28a2b2 + 21a4 = 2 ? 7 ? a ? a ? 2 ? 2 ? 7 ? a ? a ? b ? b + 3 ? 7 ? a ? a ? a ? a

= 7 ? a ? a(2 ? 4 ? b2 + 3 ? a2) = 7a2(2 ? 4b2 + 3a2) 14a2 ? 28a2b2 + 21a4 = 7a2(2 ? 4b2 + 3a2)

2

7. a4b ? 6a2b2 + 3ab3 we can write, a4b = a ? a ? a ? a ? b 6a2b2 = 2 ? 3 ? a ? a ? b ? b 3ab3 = 3 ? a ? b ? b ? b a4b ? 6a2b2 + 3ab3 = a ? a ? a ? a ? b ? 2 ? 3 ? a ? a ? b ? b + 3 ? a ? b ? b ? b = a ? b(a3 ? 6ab + 3b2) a4b ?6a2b2 + 3ab3 = ab(a3 ? 6ab + 3b2)

8. x4y4 + x4y2 ? x2y4 = x2y2(x2y2 + x2 ? y2)

9. 9x3y2 + 18xy4 ? 27x2y3 = 9xy2(x2 + 2y2 ? 3xy)

10. ?5x2 + 5xy ? 5xz = ?5x(x ? y + z)

11. 3x2y2 ? 2xy2 + 6x2y = xy(3xy ? 2y + 6x)

12. 12a3b2 + 18ab4 ? 6a2b3 = 6ab2(2a2 + 3b2 ? ab)

13. 16x2y + 12bxy = 4xy(4x + 3b) 14. 3x3 ? 6x2 + 12x = 3x(x2 ? 2x + 4) 15. 64x6y6 ? 72x4y6 = 8x4y6(8x2 ? 9) 16. 6x3y2 + 8x2y ? 2xy = 2xy(3x2y + 4x ? 1) 17. 15a3 + 45a2 + 30a = 15a(a2 + 3a + 2) 18. 18a2bc ? 12ab2c + 6abc2

= 6abc(3a ? 2b + c) 19. 8(x ? 3y)2 + 4(x ? 3y)

= 4(x ? 3y) [2(x ? 3y) + 1] [Taking 4(x ? 3y) common]

= 4(x ? 3y) (2x ? 6y + 1) 20. 3x(a ? 3b) ? y(a ? 3b)

= (a ? 3b) (3x ? y) [Taking (a ? 3b) common]

21. ab2 + ab + b + 1 = (ab2 + ab) + (b + 1) = ab(b + 1) + (b + 1) = (b + 1) (ab + 1) [Taking (b + 1) common]

22. x3 + y3 + xy (x + y) = x3 + y3 + x2y + xy2 = (x3 + x2y) + (y3 + xy2) = x2(x + y) + y2(y + x) = x2(x + y) + y2(x + y) = (x + y) (x2 + y2) [Taking (x + y) common]

23. 3xy ? y2 + 6xz ? 2yz = (3xy ? y2) + (6xz ? 2yz) = y(3x ? y) + 2z(3x ? y) = (3x ? y) (y + 2z) [Taking (3x ? y) common]

Answer Keys

24. x2y + xy + x + 1 = (x2y + xy) + (x + 1) = xy(x + 1) + (x + 1) = (x + 1) (xy + 1) [Taking (x + 1) common]

25. ab(x2 + y2) + xy (a2 + b2) = abx2 + aby2 + xya2 + xyb2 = (abx2 + xya2) + (aby2 + xyb2) = ax(bx + ay) + by(ay + bx) = ax(ay + bx) + by(ay + bx) = (ay + bx) (ax + by)

26. a2 ? a(x + 3b) + 3bx = a2 ? ax ? 3ab + 3bx = (a2 ? ax) ? (3ab ? 3bx) = a(a ? x) ? 3b(a ? x) = (a ? x) (a ? 3b) [Taking (a ? x) common]

27. (a ? b) + (a ? b)2 = (a ? b) [1 + (a ? b)] = (a ? b) (1 + a ? b)

28. (x + y) (3a ? b) ? (2x ? 3y) (3a ? b) = (3a ? b) [(x + y) ? (2x ? 3y)] = (3a ? b) [x + y ? 2x + 3y] = (3a ? b) (? x + 4y) = (3a ? b) (4y ? x)

29. 7(a + 2b)2 ? 21(a + 2b) = 7(a + 2b) [(a + 2b) ? 3] [Taking 7(a + 2b) common] = 7(a + 2b) (a + 2b ? 3)

30. 4a(5a ? 6b) ? 12a2(5a ? 6b) = 4a(5a ? 6b) (1 ? 3a)

EXERCISE 7.2

1. 9a2 ? 16b2 = (3a)2 ? (4b)2 = (3a + 4b) (3a ? 4b) [ a2 ? b2 = (a + b) (a ? b)]

2. 16x4 ? 144x2 = 16x2(x2 ? 9) = 16x2 (x2 ? 32) [Taking 16x2 common] = 16x2 (x + 3) (x ? 3) [ a2 ? b2 = (a + b) (a ? b)]

3. 3x2 ? 12 = 3(x2 ? 4) = 3(x2 ? 22) = 3(x + 2) (x ? 2) [ a2 ? b2 = (a + b) (a ? b)]

4. x2 ? 64y2 = x2 ? (8y)2 = (x + 8y) (x ? 8y) [ a2 ? b2 = (a + b) (a ? b)]

Mathematics In Everyday Life-8

5. (a + b)2 ? (a ? b)2 = [(a + b) + (a ? b)] [(a + b) ? (a ? b)] [ a2 ? b2 = (a + b) (a ? b)] = (a + b + a ? b) (a + b ? a + b) = (2a) (2b) = 4ab

6. 16a2 ? 225y2 = (4a)2 ? (15y)2 = (4a + 15y) (4a ? 15y) [ a2 ? b2 = (a + b) (a ? b)]

7. (2x + 3y)2 ? 9z2 = (2x + 3y)2 ? (3z)2 = (2x + 3y + 3z) (2x + 3y ? 3z) [ a2 ? b2 = (a + b) (a ? b)]

8.

9x2 ?

1 144

=

(3x)2

1 12

2

=

3x

1 12

3x

1 12

9. 3x5 ? 243x = 3x(x4 ? 81)

= 3x[(x2)2 ? (9)2] = 3x(x2 + 9) (x2 ? 9)

[ a2 ? b2 = (a + b) (a ? b)] = 3x(x2 + 9) (x2 ? 32) = 3x(x2 + 9)(x + 3)(x ? 3)

[ a2 ? b2 = (a + b) (a ? b)]

10. 16(x + y)2 ? 25(x ? 3y)2

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

[4(x + y) ? 5(x ? 3y)] [ a2 ? b2 = (a + b) (a ? b)] = (4x + 4y + 5x ? 15y)

(4x + 4y ? 5x + 15y) = (9x ? 11y) (?x + 19y) = ? (9x ? 11y) (x ? 19y)

11. 81 ? p4 = [92 ? (p2)2]

= (9 + p2) (9 ? p2) [ a2 ? b2 = (a + b) (a ? b)]

= (9 + p2) (32 ? p2)

= (9 + p2) (3 + p) (3 ? p) [ a2 ? b2 = (a + b) (a ? b)]

12. 3x4 ? 3 = 3(x4 ? 1)

[Taking 3 common]

= 3[(x2)2 ? (1)2]

= 3(x2 + 1) (x2 ? 1) [ a2 ? b2 = (a + b) (a ? b)]

= 3(x2 + 1) (x + 1) (x ? 1) [ a2 ? b2 = (a + b) (a ? b)]

13. (2a + 3b)2 ? 25c2 = (2a + 3b)2 ? (5c)2 = (2a + 3b + 5c) (2a + 3b ? 5c) [ a2 ? b2 = (a + b) (a ? b)]

14. 3 ? 75x2 = 3(1 ? 25x2) = 3[1 ? (5x)2] = 3(1 + 5x) (1 ? 5x)

3

15. x4 ? 16y4 = [(x2)2 ?(4y2)2] = (x2 + 4y2) (x2 ? 4y2) [ a2 ? b2 = (a + b) (a ? b)] = (x2 + 4y2) [x2 ? (2y)2] = (x2 + 4y2) (x + 2y) (x ? 2y) [ a2 ? b2 = (a + b) (a ? b)]

16. x12y4 ? x4y12 = x4y4(x8 ? y8) = x4y4[(x4)2 ? (y4)2] = x4y4(x4 + y4) (x4 ? y4) [ a2 ? b2 = (a + b) (a ? b)] = x4y4 (x4 + y4) [(x2)2 ? (y2)2] = x4y4(x4 + y4) (x2 + y2) (x2 ? y2) [ a2 ? b2 = (a + b) (a ? b)] = x4y4(x4 + y4) (x2 + y2) (x + y) (x ? y) [ a2 ? b2 = (a + b) (a ? b)]

17.

16a2

25 9a2

= (4a)2

5 3a

2

=

4a

5 3a

4a

5 3a

[ a2 ? b2 = (a + b) (a ? b)]

18. (x + 2)2 ? (x + 4)2

= [(x + 2) + (x + 4)] [(x + 2) ? (x + 4)]

= (x + 2 + x + 4) (x + 2 ? x ? 4)

= (2x + 6)(? 2)

= ? 2 ? 2 (x + 3)

= ?4(x + 3)

19. (2 + 3a)2 ? 36a2 = (2 + 3a)2 ? (6a)2

= (2 + 3a + 6a) (2 + 3a ? 6a) [ a2 ? b2 = (a + b) (a ? b)]

= (2 + 9a) (2 ? 3a)

20. x3 ? 225x = x(x2 ? 225)

[Taking x common]

= x[x2 ? (15)2]

= x(x + 15) (x ? 15)

21. (3x ? 4y)4 ? x4

= [(3x ? 4y)2]2 ? (x2)2

= [(3x ? 4y)2 + x2] [(3x ? 4y)2 ? x2] [ a2 ? b2 = (a + b) (a ? b)]

= [(3x ? 4y)2 + x2] (3x ? 4y + x) (3x ? 4y ? x) [ a2 ? b2 = (a + b) (a ? b)]

= [9x2 + 16y2 ? 24xy + x2] (4x ? 4y) (2x ? 4y)

a2 b2 (a b) (a ? b)

and

(a

b)2

(a2

2ab

b2

)

= (10x2 + 16y2 ? 24xy) (4x ? 4y) (2x ? 4y)

= 2 ? 4 ? 2(5x2 + 8y2 ? 12xy) (x ? y) (x ? 2y)

= 16(5x2 + 8y2 ? 12xy) (x ? y) (x ? 2y)

4

22. (a ? b)2 ? (x ? y)2

= [(a ? b) + (x ? y)] [(a ? b) ? (x ? y)] [ a2 ? b2 = (a + b) (a ? b)]

= (a ? b + x ? y) (a ? b ? x + y) 23. x8 ? 1 = (x4)2 ? (1)2

= (x4 +1) (x4 ? 1) [ a2 ? b2 = (a + b) (a ? b)]

= (x4 + 1) [(x2)2 ? (1)2] = (x4 + 1) (x2 + 1) (x2 ? 1)

[ a2 ? b2 = (a + b) (a ? b)] = (x4 + 1) (x2 + 1) (x + 1) (x ? 1)

[ a2 ? b2 = (a + b) (a ? b)] 24. 169a2 ? 144b2 = (13a)2 ? (12b)2

= (13a + 12b) (13a ? 12b) [ a2 ? b2 = (a + b) (a ? b)]

25. x5 ? 81x3 = x3(x2 ? 81) = x3(x2 ? 92) = x3(x + 9)(x ? 9) [ a2 ? b2 = (a + b) (a ? b)]

26.

1 x2y2 1 y2z2

9

16

=

y2

1

9

x2

?

1 16

z

2

=

y2

1 3

x

2

1 4

z

2

=

y2

1 3

x

1 4

z

1 3

x

1 4

z

[ a2 ? b2 = (a + b) (a ? b)]

=

y2

x 3

z 4

x 3

z 4

27. 3a5 ? 48a3 = 3a3(a2 ? 16)

= 3a3(a2 ? 42)

= 3a3 (a + 4) (a ? 4) [ a2 ? b2 = (a + b)(a ? b)]

28. (2x ? y)2 ? 9z2

= (2x ? y)2 ? (3z)2

= [(2x ? y) + 3z] [(2x ? y) ? 3z]

= (2x ? y + 3z) (2x ? y ? 3z) [ a2 ? b2 = (a + b) (a ? b)]

29. 125a2 ? 45b2 = 5(25a2 ? 9b2)

[Taking 5 common]

= 5[(5a)2 ? (3b)2]

= 5(5a + 3b) (5a ? 3b)

[ a2 ? b2 = (a + b) (a ? b)]

30. 3x3y ? 243 xy3 = 3xy(x2 ? 81y2) [Taking 3xy common]

= 3xy[x2 ? (9y)2]

= 3xy(x + 9y)(x ? 9y) [ a2 ? b2 = (a + b) (a ? b)]

Answer Keys

31. 4 ? 16(a ? b)2 = 4[1 ? 4(a ? b)2] = 4[(1)2 ? {2(a ? b)}2] = 4[1 + 2(a ? b)] [1 ? 2(a ? b)] [ a2 ? b2 = (a + b) (a ? b)] = 4(1 + 2a ? 2b) (1 ? 2a + 2b)

32. 9a2 ? 4b2 ? 3a ? 2b = [(3a)2 ? (2b)2] ? (3a + 2b) = (3a + 2b) (3a ? 2b) ? (3a + 2b) [ a2 ? b2 = (a + b) (a ? b)] = (3a + 2b) (3a ? 2b ? 1) [Taking (3a + 2b) common]

33. x2 ? y2 ? x ? y = (x2 ? y2) ? (x + y) = (x + y) (x ? y) ? (x + y) = (x + y) (x ? y ? 1)

34. x2 ? 3x + 3y ? y2 = (x2 ? y2) ? 3(x ? y) = (x + y) (x ? y) ? 3(x ? y) [ a2 ? b2 = (a + b) (a ? b)] = (x ? y) (x + y ? 3)

35. x7y3 ? x3y7 = x3y3 (x4 ? y4) = x3y3[(x2)2 ? (y2)2] = x3y3(x2 + y2) (x2 ? y2) = x3y3(x2 + y2) (x + y) (x ? y) [ a2 ? b2 = (a + b) (a ? b)]

36. a(a + c) ? b(b + c) = a2 + ac ? b2 ? bc = (a2 ? b2) + ac ? bc = (a2 ? b2) + c(a ? b) = (a + b) (a ? b) + c(a ? b) = (a ? b) [(a + b) + c] [Taking (a ? b) common] = (a ? b) (a + b + c)

37. (i) (205)2 ? (195)2 = (205 + 195) (205 ? 195) [ a2 ? b2 = (a + b) (a ? b)]

= 400 ? 10 = 4000 (ii) (7.6)2 ? (2.4)2 = (7.6 + 2.4)(7.6 ? 2.4)

[ a2 ? b2 = (a + b) (a ? b)] = 10 ? 5.2 = 52 (iii) (501)2 ? (499)2 = (501 + 499) (501 ? 499) = [ a2 ? b2 = (a + b) (a ? b)] = 1000 ? 2 = 2000

Mathematics In Everyday Life-8

EXERCISE 7.3

1. 4x2 ? 12x + 9 = (2x)2 ? 2(2x)(3) + (3)2 = (2x ? 3)2 [ (a ? b)2 = a2 ? 2ab + b2]

2. a2 ? 8a + 16 = a2 ? 2(a)(4) + (4)2 = (a ? 4)2 [ (a ? b)2 = a2 ? 2ab + b2]

3. x2 ? 14x + 49 = x2 ? 2x(7) + (7)2 = (x ? 7)2 [ (a ? b)2 = a2 ? 2ab + b2]

4. 25a2 + 30a + 9 = (5a)2 + 2(5a) (3) + 32 = (5a + 3)2 [ (a + b)2 = a2 + 2ab + b2]

5. 49m2 + 84mn + 36n2 = (7m)2 + 2(7m)(6n) + (6n)2 = (7m + 6n)2 [ (a + b)2 = a2 + 2ab + b2]

6. 16a2 ? 24a + 9 = (4a)2 ? 2(4a) ? (3) + 32 = (4a ? 3)2 [ (a ? b)2 = a2 ? 2ab + b2]

7. 4x2 + 4x + 1 = (2x)2 + 2(2x) ? 1 + (1)2 = (2x + 1)2 [ (a + b)2 = a2 + 2ab + b2]

8. 9x2 + 30xy + 25y2 = (3x)2 + 2(3x)(5y) + (5y)2 = (3x + 5y2) [ (a + b)2 = a2 + 2ab + b2]

9. m2 ? 8mn + 16n2 = m2 ? 2m(4n) + (4n)2 = (m ? 4n)2 [ (a ? b)2 = a2 ? 2ab + b2]

10. 4x2 ? 12xy + 9y2 = (2x)2 ? 2(2x)(3y) + (3y)2 = (2x ? 3y)2 [ (a ? b)2 = a2 ? 2ab + b2]

11. 4x2 + 20xy + 25y2 = (2x)2 + 2(2x)(5y) + (5y)2 = (2x + 5y)2 [ (a + b)2 = a2 + 2ab + b2]

12. 9x2 + 42xy + 49y2 = (3x)2 + 2(3x)(7y) + (7y)2 = (3x + 7y)2 [ (a + b)2 = a2 + 2ab + b2]

13.

4x2 ?

4 x 1 39

= (2x)2 ?

2

(2x)

1 3

1 3

2

=

2x

1 3

2

[ (a ? b)2 = a2 ? 2ab + b2]

5

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