Chapter 5 : Factorisation - Frank Solutions for Class 9 Maths ICSE

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Chapter 5 - Factorisation Excercise Ex. 5.1

Question 1

Factorise the following by taking out the common factors:

4x2y3 - 6x3y2 - 12xy2

Solution 1

Question 2

Factorise the following by taking out the common factors:

5a(x2 - y2) + 35b(x2 - y2)

Solution 2

Question 3

Factorise the following by taking out the common factors:

2x5y + 8x3y2 - 12x2y3

Solution 3

Question 4

Factorise the following by taking out the common factors:

12a3 + 15a2b - 21ab2

Solution 4

Question 5

Factorise the following by taking out the common factors:

24m4n6 + 56m6n4 - 72m2n2

Solution 5

Question 6

Factorise the following by taking out the common factors:

(a - b)2 -2(a - b)

Solution 6

Question 7

Factorise the following by taking out the common factors:

2a(p2 + q2) + 4b(p2 + q2)

Solution 7

Question 8

Factorise the following by taking out the common factors:

81(p + q)2 -9p - 9q

Solution 8

Question 9

Factorise the following by taking out the common factors:

(mx + ny)2 + (nx - my)2

Solution 9

Question 10

Factorise the following by taking out the common factors:

36(x + y)3 - 54(x + y)2

Solution 10

Question 11

Factorise the following by taking out the common factors:

p(p2 + q2 - r2) + q(r2 - q2 -p2) - r(p2 + q2 - r2)

Solution 11

Question 12

Factorise the following by grouping the terms:

15xy - 9x - 25y + 15

Solution 12

15xy - 9x - 25y + 15

= (15xy - 9x) - (25y + 15)

= 3x(5y - 3) - 5(5y - 3)

= (5y - 3)(3x - 5)

Question 13

Factorise the following by grouping the terms:

15x2 + 7y - 3x - 35xy

Solution 13

15x2 + 7y - 3x - 35xy

= 15x2 - 3x - 35xy + 7y

= (15x2 - 3x) - (35xy - 7y)

= 3x(5x - 1) - 7y(5x - 1)

= (5x - 1)(3x - 7y)

Question 14

Factorise the following by grouping the terms:

9 + 3xy + x2y + 3x

Solution 14

9 + 3xy + x2y + 3x

= 9 + 3xy + 3x + x2y

= (9 + 3xy) + (3x + x2y)

= 3(3 + xy) + y(3 + xy)

= (3 + xy)(3 + x)

Question 15

Factorise the following by grouping the terms:

8(2a + b)2 - 8a -4b

Solution 15

8(2a + b)2 - 8a - 4b

= 8(2a + b)2 - (8a + 4b)

= 8(2a + b)2 - 4(2a + b)

= 4(2a + b)[2(2a + b) - 1]

= 4(2a + b)[4a + 2b - 1]

Question 16

Factorise the following by grouping the terms:

x(x - 4)- x + 4

Solution 16

x(x - 4) - x + 4

= x(x - 4) - 1(x - 4)

= (x - 4)(x - 1)

Question 17

Factorise the following by grouping the terms:

2m3 - 5n2 - 5m2n + 2mn

Solution 17

2m3 - 5n2 - 5m2n + 2mn

= 2m3 + 2mn - 5m2n - 5n2

= (2m3 + 2mn) - (5m2n + 5n2)

= 2m(m2 + n) - 5n(m2 + n)

= (m2 + n)(2m - 5n)

Question 18

Factorise the following by grouping the terms:

4abx2 + 49aby2 + 14xy(a2 + b2)

Solution 18

4abx2 + 49aby2 + 14xy(a2 + b2)

= 4abx2 + 49aby2 + 14a2xy + 14b2xy

= (4abx2 + 14a2xy) + (14b2xy + 49aby2)

= 2ax(2bx + 7ay) + 7by(2bx + 7ay)

= (2bx + 7ay)(2ax + 7by)

Question 19

Factorise the following by grouping the terms:

9x3 + 6x2y2 - 4y3 - 6xy

Solution 19

9x3 + 6x2y2 - 4y3 - 6xy

= 9x3 + 6x2y2 - 6xy - 4y3

= (9x3 + 6x2y2) - (6xy + 4y3)

= 3x2(3x + 2y2) - 2y(3x + 2y2)

= (3x + 2y2)(3x2 - 2y)

Question 20

Factorise the following by grouping the terms:

3ax2 - 5bx2 + 9az2 + 6ay2 - 10by2 - 15bz2

Solution 20

3ax2 - 5bx2 + 9az2 + 6ay2 - 10by2 - 15bz2

= 3ax2 + 6ay2 + 9az2 - 5bx2 - 10by2 - 15bz2

= (3ax2 + 6ay2 + 9az2) - (5bx2 + 10by2 + 15bz2)

= 3a(x2 + 2y2 + 3z2) - 5b(x2 + 2y2 + 3z2)

= (x2 + 2y2 + 3z2)(3a - 5b)

Question 21

Factorise the following by grouping the terms:

8x3 - 24x2y + 54xy2 -162y3

Solution 21

8x3 - 24x2y + 54xy2 - 162y3

= (8x3 - 24x2y) + (54xy2 - 162y3)

= 8x2(x - 3y) + 54y2(x - 3y)

= (x - 3y)(8x2 + 54y2)

Question 22

Factorise the following by grouping the terms:

2a + b + 3c - d + (2a + b)3 + (2a + b)2(3c - d)

Solution 22

2a + b + 3c - d + (2a + b)3 + (2a + b)2(3c - d)

= (2a + b + 3c - d) + [(2a + b)3 + (2a + b)2(3c - d)]

= 1(2a + b + 3c - d) + (2a + b)2(2a + b + 3c - d)

= (2a + b + 3c - d)[1 + (2a + b)2]

Question 23

Factorise the following by grouping the terms:

xy(a2 + 1) + a(x2 + y2)

Solution 23

xy(a2 + 1) + a(x2 + y2)

= a2xy + xy + ax2 + ay2

= (a2xy + ax2) + (ay2 + xy)

= ax(ay + x) + y(ay + x)

= (ay + x)(ax + y)

Question 24

Factorise the following by grouping the terms:

p2x2 + (px2 + 1)x + p

Solution 24

p2x2 + (px2 + 1)x + p

= p2x2 + px3 + x + p

= (p2x2 + px3) + (p + x)

= px2(p + x) + 1(p + x)

= (p + x)(px2 + 1)

Question 25

Factorise the following by grouping the terms:

x2 - (p + q)x + pq

Solution 25

x2 - (p + q)x + pq

= x2 - px - qx + pq

= (x2 - px) - (qx + pq)

= x(x - p) - q(x - p)

= (x - p)(x - q)

Question 26

Factorise the following by grouping the terms:

Solution 26

Question 27

Factorise the following by grouping the terms:

x + y + m(x + y)

Solution 27

x + y + m(x + y)

= (x + y) + m(x + y)

= (x + y)(1 + m)

Question 28

Factorise the following by grouping the terms:

 

Solution 28

Question 29

Factorise the following by grouping the terms:

2p(a2 - 2b2) -14p + (a2 - 2b2)2 - 7(a2 - 2b2)

Solution 29

2p(a2 - 2b2) - 14p + (a2 - 2b2)2 - 7(a2 - 2b2)

= 2p(a2 - 2b2) + (a2 - 2b2)2 - 14p - 7(a2 - 2b2)

= [2p(a2 - 2b2) + (a2 - 2b2)2] - [14p + 7(a2 - 2b2)]

= (a2 - 2b2)(2p + a2 - 2b2) - 7(2p + a2 - 2b2)

= (2p + a2 - 2b2)(a2 - 2b2 - 7)

Chapter 5 - Factorisation Excercise Ex. 5.2

Question 1

Factorise the following by splitting the middle term:

x2 + 6x + 8

Solution 1

x2 + 6x + 8

= x2 + 4x + 2x + 8

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

= (x + 4)(x + 2)

Question 2

Factorise the following by splitting the middle term:

x2 - 11x + 24

Solution 2

x2 - 11x + 24

= x2 - 8x - 3x + 24

= x(x - 8) - 3(x - 8)

= (x - 8)(x - 3)

Question 3

Factorise the following by splitting the middle term:

x2 + 5x - 6

Solution 3

x2 + 5x - 6

= x2 + 6x - x - 6

= x(x + 6) - 1(x + 6)

= (x + 6)(x - 1)

Question 4

Factorise the following by splitting the middle term:

p2- 12p - 64

Solution 4

p2 - 12p - 64

= p2 - 16p + 4p - 64

= p(p - 16) + 4(p - 16)

= (p - 16)(p + 4)

Question 5

Factorise the following by splitting the middle term:

y2 - 2y - 24

Solution 5

y2 - 2y - 24

= y2 - 6y + 4y - 24

= y(y - 6) + 4(y - 6)

=(y - 6)(y + 4)

Question 6

Factorise the following by splitting the middle term:

3x2 + 19x - 14

Solution 6

3x2 + 19x - 14

= 3x2 + 21x - 2x - 14

= 3x(x + 7) - 2(x + 7)

= (x + 7)(3x - 2)

Question 7

Factorise the following by splitting the middle term:

15a2 - 14a - 16

Solution 7

15a2 - 14a - 16

= 15a2 - 24a + 10a - 16

= 3a(5a - 8) + 2(5a - 8)

= (5a - 8)(3a + 2)

Question 8

Factorise the following by splitting the middle term:

12 + x - 6x2

Solution 8

12 + x - 6x2

= 12 + 9x - 8x - 6x2

= 3(4 + 3x) - 2x(4 + 3x)

= (4 + 3x)(3 - 2x)

Question 9

Factorise the following by splitting the middle term:

7x2 + 40x - 12

Solution 9

7x2 + 40x - 12

= 7x2 + 42x - 2x - 12

= 7x(x + 6) - 2(x + 6)

= (x + 6)(7x - 2)

Question 10

Factorise the following:

5x2 - 17xy + 6y2

Solution 10

5x2 - 17xy + 6y2

= 5x2 - 15xy - 2xy + 6y2

= 5x(x - 3y) - 2y(x - 3y)

= (x - 3y)(5x - 2y)

Question 11

Factorise the following:

9x2 - 22xy + 8y2

Solution 11

9x2 - 22xy + 8y2

= 9x2 - 18xy - 4xy + 8y2

= 9x(x - 2y) - 4y(x - 2y)

= (x - 2y)(9x - 4y)

Question 12

Factorise the following:

2x3 + 5x2y - 12xy2

Solution 12

2x3 + 5x2y - 12xy2

= 2x3 + 8x2y - 3x2y - 12xy2

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

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

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

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

Question 13

Factorise the following:

x2y2 + 15xy - 16

Solution 13

x2y2 + 15xy - 16

= x2y2 + 16xy - xy - 16

= xy(xy + 16) - 1(xy + 16)

= (xy + 16)(xy - 1)

Question 14

Factorise the following:

(2p + q)2 - 10p - 5q - 6

Solution 14

(2p + q)2 - 10p - 5q - 6

= (2p + q)2 - (10p - 5q) - 6

= (2p + q)2 - 5(2p + q) - 6

= (2p + q)2 - 6(2p + q) + (2p + q) - 6

= (2p + q)(2p + q - 6) + 1(2p + q - 6)

= (2p + q - 6)(2p + q + 1) 

Question 15

Factorise the following:

y2 + 3y + 2 + by + 2b

Solution 15

y2 + 3y + 2 + by + 2b

= y2 + y + 2y + 2 + by + 2b

= y2 + y + by + 2y + 2 + 2b

= y(y + 1 + b) + 2(y + 1 + b)

= (y + 1 + b)(y + 2)

Question 16

Factorise the following:

x3y3 - 8x2y2 + 15xy

Solution 16

x3y3 - 8x2y2 + 15xy

= x3y3 - 3x2y2 - 5x2y2 + 15xy

= x2y2(xy - 3) - 5xy(xy - 3)

= (xy - 3)(x2y2 - 5xy)

= (xy - 3)xy(xy - 5)

= xy(xy - 3)(xy - 5)

Question 17

Factorise the following:

Solution 17

Question 18

Factorise the following:

Solution 18

Question 19

Factorise the following:

5(3x + y)2 + 6(3x + y) - 8

Solution 19

5(3x + y)2 + 6(3x + y) - 8

= 5(3x + y)2 + 10(3x + y) - 4(3x + y) - 8

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

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

 

Question 20

Factorise the following:

5 - 4(a - b) - 12(a - b)2

Solution 20

5 - 4(a - b) - 12(a - b)2

= 5 - 10(a - b) + 6(a - b) - 12(a - b)2

= 5[1 - 2(a - b)] + 6(a - b)[1 - 2(a - b)]

= [5 + 6(a - b)][1 - 2(a - b)]

= (5 + 6a - 6b)(1 - 2a + 2b)

Question 21

Factorise the following:

(3a - 2b)2 +3(3a - 2b) - 10

Solution 21

(3a - 2b)2 + 3(3a - 2b) - 10

= (3a - 2b)2 + 5(3a - 2b) - 2(3a - 2b) - 10

= (3a - 2b)(3a - 2b + 5) - 2(3a - 2b +5)

= (3a - 2b + 5)((3a - 2b - 2)

Question 22

Factorise the following:

(a2 - 2a)2 - 23(a2 - 2a) + 120

Solution 22

(a2 - 2a)2 - 23(a2 - 2a) + 120

= (a2 - 2a)2 - 15(a2 - 2a) - 8(a2 - 2a) + 120

= (a2 - 2a)(a2 - 2a - 15) - 8(a2 - 2a - 15)

= (a2 - 2a - 15)(a2 - 2a - 8)

= (a2 - 5a + 3a - 15)(a2 - 4a + 2a - 8)

= [a(a - 5) + 3(a - 5)][a(a - 4) + 2(a - 4)]

= [(a - 5)(a + 3)][(a - 4)(a + 2)]

= (a - 5)(a + 3)(a - 4)(a + 2)

= (a + 2)(a + 3)(a - 4)(a - 5)

Question 23

Factorise the following:

(x + 4)2 - 5xy - 20y - 6y2

Solution 23

(x + 4)2 - 5xy - 20y - 6y2

= (x + 4)2 - 5y(x + 4) - 6y2

= (x + 4)2 - 6y(x + 4) + y(x + 4) - 6y2

= (x + 4)(x + 4 - 6y) + y(x + 4 - 6y)

= (x + 4 - 6y)(x + 4 + y)

= (x - 6y + 4)(x + y + 4)

Question 24

Factorise the following:

7(x - 2)2 - 13(x - 2) - 2

Solution 24

7(x - 2)2 - 13(x - 2) - 2

= 7(x - 2)2 - 14(x - 2) + (x - 2) - 2

= 7(x - 2)(x - 2 - 2) + 1(x - 2 - 2)

= 7(x - 2)(x - 4) + 1(x - 4)

= (x - 4)[7(x - 2) + 1]

= (x - 4)(7x - 14 + 1)

= (x - 4)(7x - 13)

Question 25

Factorise the following:

12 - (y + y2)(8 - y - y2)

Solution 25

12 - (y + y2)(8 - y - y2)

= 12 - a(8 - a) [Taking y + y2 = a]

= 12 - 8a + a2

= 12 - 6a - 2a + a2

= 6(2 - a) - a(2 - a)

= (2 - a)(6 - a)

= [2 - (y + y2)][6 - (y + y2)]

= (2 - y - y2)(6 - y - y2)

= (2 - 2y + y - y2)(6 - 3y + 2y - y2)

= [2(1 - y) + y(1 - y)][3(2 - y) + y(2 - y)]

= [(1 - y)(2 + y)][(2 - y)(3 + y)]

= (1 - y)(2 + y)(2 - y)(3 + y)

= (y - 1)(y + 2)(y - 2)(y + 3)

Question 26

Factorise the following:

(p2 + p)2 - 8(p2 + p) + 12

Solution 26

(p2 + p)2 - 8(p2 + p) + 12

= (p2 + p)2 - 6(p2 + p) - 2(p2 + p) + 12

= (p2 + p)(p2 + p - 6) - 2(p2 + p - 6)

= (p2 + p - 6)(p2 + p - 2)

= (p2 + 3p - 2p - 6)(p2 + 2p - p - 2)

= [p(p + 3) - 2(p + 3)][p(p + 2) - 1(p + 2)]

= [(p + 3)(p - 2)][(p + 2)(p - 1)]

= (p + 3)(p - 2)(p + 2)(p - 1)

Question 27

Factorise the following:

(y2 - 3y)(y2 - 3y + 7) + 10

Solution 27

(y2 - 3y)(y2 - 3y + 7) + 10

 = a(a + 7) + 10 [taking (y2 - 3y) = a]

= a2 + 7a + 10

= a2 + 5a + 2a + 10

= a(a + 5) + 2(a + 5)

= (a + 5)(a + 2)

= (y2 - 3y + 5)(y2 - 3y + 2)

= (y2 - 3y + 5)(y2 - 2y - y + 2)

= (y2 - 3y + 5)[y(y - 2) - 1(y - 2)]

= (y2 - 3y + 5)[(y - 2)(y - 1)]

= (y - 1)(y - 2)(y2 - 3y + 5)

Question 28

Factorise the following:

(t2 - t)(4t2 - 4t - 5) - 6

Solution 28

(t2 - t)(4t2 - 4t - 5) - 6

= (t2 - t)[4(t2 - t) - 5] - 6

= a[4a - 5] - 6 [Taking (t2 - t) = a]

= 4a2 - 5a - 6

= 4a2 - 8a + 3a - 6

= 4a(a - 2) + 3(a - 2)

= (a - 2)(4a + 3)

= (t2 - t - 2)[4(t2 - t) + 3]

= (t2 - 2t + t - 2)(4t2 - 4t + 3)

= [t(t - 2) + 1(t - 2)](4t2 - 4t + 3)

= [(t - 2)(t + 1)](4t2 - 4t + 3)

= (t + 1)(t - 2)(4t2 - 4t + 3)

Question 29

Factorise the following:

12(2x - 3y)2 - 2x + 3y - 1

Solution 29

12(2x - 3y)2 - 1(2x - 3y) - 1

= 12a2 - a - 1 [Taking (2x - 3y) = a]

= 12a2 - 4a + 3a - 1

= 4a(3a - 1) + 1(3a - 1)

= (3a - 1)(4a + 1)

= [3(2x - 3y) - 1][4(2x - 3y) + 1]

= (6x - 9y - 1)(8x - 12y + 1)

Question 30

Factorise the following:

6 - 5x + 5y + (x - y)2

Solution 30

6 - 5x + 5y + (x - y)2

= 6 - 5(x - y) + (x - y)2

= 6 - 3(x - y) - 2(x - y) + (x - y)2

= 3[2 - (x - y)] - (x - y)[2 - (x - y)]

= 3(2 - x + y) - (x - y)(2 - x + y)

= (2 - x + y)(3 - x + y)

Question 31

Factorise the following:

Solution 31

Question 32

Factorise the following:

p4 + 23p2q2 + 90q4

Solution 32

P4 + 23p2q2 + 90q4

= p4 + 18p2q2 + 5p2q2 + 90q4

= p2(p2 + 18q2) + 5q2(p2 + 18q2)

= (p2 + 18q2)(p2 + 5q2)

Question 33

Factorise the following:

2a3 + 5a2b - 12ab2

Solution 33

2a3 + 5a2b - 12ab2

= 2a3 + 8a2b - 3a2b - 12ab2

= 2a2(a + 4b) - 3ab(a + 4b)

= (a + 4b)(2a2 - 3ab)

= (a + 4b)a(2a - 3b)

= a(a + 4b)(2a - 3b)

Chapter 5 - Factorisation Excercise Ex. 5.3

Question 1

Factorise the following by the difference of two squares:

x2 - 16

Solution 1

x2 - 16

= x2 - 42

= (x - 4)(x + 4)

Question 2

Factorise the following by the difference of two squares:

64x2 - 121y2

Solution 2

64x2 - 121y2

= (8x)2 - (11y)2

= (8x - 11y)(8x + 11y)

Question 3

Factorise the following by the difference of two squares:

441 - 81y2

Solution 3

441 - 81y2

= (21)2 - (9y)2

= (21 - 9y)(21 + 9y)

= 3(7 - 3y)3(7 + 3y)

= 9(7 - 3y)(7 + 3y)

Question 4

Factorise the following by the difference of two squares:

x6 - 196

Solution 4

x6 - 196

= (x3)2 - (14)2

= (x3 - 14)(x3 + 14)

Question 5

Factorise the following by the difference of two squares:

625 - b2

Solution 5

625 - b2

= (25)2 - (b)2

= (25 - b)(25 + b)

Question 6

Factorise the following by the difference of two squares:

Solution 6

Question 7

Factorise the following by the difference of two squares:

8xy2 - 18x3

Solution 7

8xy2 - 18x3

= 2x(4y2 - 9x2)

= 2x[(2y)2 - (3x)2]

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

= 2x(2y - 3x)(2y + 3x)

Question 8

Factorise the following by the difference of two squares:

16a4 - 81b4

Solution 8

16a4 - 81b4

= (4a2)2 - (9b2)2

= (4a2 - 9b2)(4a2 + 9b2)

= [(2a)2 - (3b)2](4a2 + 9b2)

= [(2a - 3b)(2a + 3b)](4a2 + 9b2)

= (2a - 3b)(2a + 3b)(4a2 + 9b2)

Question 9

Factorise the following by the difference of two squares:

a(a - 1) - b(b - 1)

Solution 9

a(a - 1) - b(b - 1)

= a2 - a - b2 + b

= a2 - b2 - a + b

= (a2 - b2) - (a - b)

= (a - b)(a + b) - (a - b)

= (a - b)(a + b - 1)

Question 10

Factorise the following by the difference of two squares:

(x + y)2 -1

Solution 10

(x + y)2 - 1

= (x + y)2 - (1)2

= (x + y + 1)(x + y - 1)

Question 11

Factorise the following by the difference of two squares:

x2 + y2 - z2 - 2xy

Solution 11

x2 + y2 - z2 - 2xy

= x2 + y2 - 2xy - z2

= (x2 + y2 - 2xy) - z2

= (x - y)2 - (z)2

= (x - y - z)(x - y + z)

Question 12

Factorise the following by the difference of two squares:

(x - 2y)2 -z2

Solution 12

(x - 2y)2 - z2

= (x - 2y)2 - (z)2

= (x - 2y - z)(x - 2y + z)

Question 13

Factorise the following:

9(a - b)2 - (a + b)2

Solution 13

9(a - b)2 - (a + b)2

= [3(a - b)]2 - (a + b)2

= [3(a - b) - (a + b)][3(a - b) + (a + b)]

= (3a - 3b - a - b)(3a - 3b + a + b)

= (2a - 4b)(4a - 2b)

= 2(a - 2b)2(2a - b)

= 4(a - 2b)(2a - b)

Question 14

Factorise the following:

25(x - y)2 - 49(c - d)2

Solution 14

25(x - y)2 - 49(c - d)2

= [5(x - y)]2 - [7(c - d)]2

= [5(x - y) - 7(c - d)][5(x - y) + 7(c - d)]

= (5x - 5y - 7c + 7d)(5x - 5y + 7c - 7d)

Question 15

Factorise the following:

(2a - b)2 -9(3c - d)2

Solution 15

(2a - b)2 - 9(3c - d)2

= (2a - b)2 - [3(3c - d)]2

= [(2a - b) - 3(3c - d)][(2a - b) + 3(3c - d)]

= (2a - b - 9c + 3d)(2a - b + 9c - 3d)

Question 16

Factorise the following:

b2 - 2bc + c2 - a2

Solution 16

b2 - 2bc + c2 - a2

= (b2 - 2bc + c2) - a2

= (b - c)2 - (a)2

= (b - c - a)(b - c + a)

Question 17

Factorise the following:

Solution 17

Question 18

Factorise the following:

(x2 + y2 - z2)2 - 4x2y2

Solution 18

(x2 + y2 - z2)2 - 4x2y2

= (x2 + y2 - z2)2 - (2xy)2

= (x2 + y2 - z2 - 2xy)(x2 + y2 - z2 + 2xy)

= [(x2 + y2 - 2xy) - z2][(x2 + y2 + 2xy) - z2]

= [(x - y)2 - z2][(x + y)2 - z2]

= [(x - y - z)(x - y + z)][(x + y - z)(x + y + z)]

= (x - y - z)(x - y + z)(x + y - z)(x + y + z)

Question 19

Factorise the following:

a2 + b2 - c2 - d2 + 2ab - 2cd

Solution 19

a2 + b2 - c2 - d2 + 2ab - 2cd

= (a2 + b2 + 2ab) - (c2 + d2 + 2cd)

= (a + b)2 - (c + d)2

= (a + b + c + d)(a + b - c - d)

Question 20

Factorise the following:

4xy - x2 - 4y2 + z2

Solution 20

4xy - x2 - 4y2 + z2

= z2 - x2 - 4y2 + 4xy

= z2 - (x2 + 4y2 - 4xy)

= z2 - (x - 2y)2

= [z - (x - 2y)][z + (x - 2y)]

= (z - x + 2y)(z + x - 2y)

Question 21

Factorise the following:

4x2 - 12ax - y2 - z2 - 2yz + 9a2

Solution 21

4x2 - 12ax - y2 - z2 - 2yz + 9a2

= (4x2 - 12ax + 9a2) - (y2 + z2 + 2yz)

= (2x - 3a)2 - (y + z)2

= [(2x - 3a) + (y + z)][(2x - 3a) - (y + z)]

= (2x - 3a + y + z)(2x - 3a - y - z)

Question 22

Factorise the following:

(x + y)3 - x - y

Solution 22

(x + y)3 - x - y

= (x + y)(x + y)2 - (x + y)

= (x + y)[(x + y)2 - 1]

= (x + y)[(x + y + 1)(x + y - 1)]

= (x + y)(x + y + 1)(x + y - 1)

Question 23

Factorise the following:

y4 + y2 + 1

Solution 23

y4 + y2 + 1

= y4 + 2y2 + 1 - y2

= (y2 + 1)2 - y2

= (y2 + 1 + y)(y2 + 1 - y)

Question 24

Factorise the following:

(a2 - b2)(c2 - d2) - 4abcd

Solution 24

(a2 - b2)(c2 - d2) - 4abcd

= a2c2 - a2d2 - b2c2 + b2d2 - 4abcd

= a2c2 + b2d2 - 2abcd - a2d2 - b2c2 - 2abcd

= (a2c2 + b2d2 - 2abcd) - (a2d2 + b2c2 + 2abcd)

= (ac - bd)2 - (ad + bc)2

= [(ac - bd) + (ad + bc)][(ac - bd) - (ad + bc)]

= (ac - bd + ad + bc)(ac - bd - ad - bc)

Question 25

Express each of the following as the difference of two squares:

(x2 - 2x + 3)(x2 + 2x + 3)

Solution 25

Question 26

Express each of the following as the difference of two squares:

(x2 - 2x + 3) (x2 - 2x - 3)

Solution 26

Question 27

Express each of the following as the difference of two squares:

(x2 + 2x - 3) (x2 - 2x + 3)

Solution 27

Question 28

Factorise:

 

Solution 28

Question 29

Factorise:

Solution 29

Question 30

Factorise:

x4 + y4 - 6x2y2

Solution 30

Question 31

Factorise:

4x4 + 25y4 + 19x2y2

Solution 31

Question 32

Factorise:

Solution 32

Question 33

Factorise:

5x2 - y2 - 4xy + 3x - 3y

Solution 33

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