# FRANK Solutions for Class 9 Maths Chapter 5 - Factorisation

Achieve high marks with the support of Frank Solutions for ICSE Class 9 Mathematics Chapter 5 Factorisation. Prepared by TopperLearning’s experts, the solutions will help you understand how to factorise mathematical expressions by removing the common factors.

Also, learn the method of using the difference of two squares to factorise the given data. Explore ICSE Class 9 Maths Frank textbook solutions any time online on our online education portal. Additionally, with the help of our concept videos and online practice tests designed by subject experts, you can get ahead in your Maths learning.

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## Chapter 5 - Factorisation Exercise Ex. 5.1

Question 1(a)

Factorise the following by taking out the common factors:

4x2y3 - 6x3y2 - 12xy2

Solution 1(a)

Question 1(b)

Factorise the following by taking out the common factors:

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

Solution 1(b)

Question 1(c)

Factorise the following by taking out the common factors:

2x5y + 8x3y2 - 12x2y3

Solution 1(c)

Question 1(d)

Factorise the following by taking out the common factors:

12a3 + 15a2b - 21ab2

Solution 1(d)

Question 1(e)

Factorise the following by taking out the common factors:

24m4n6 + 56m6n4 - 72m2n2

Solution 1(e)

Question 1(f)

Factorise the following by taking out the common factors:

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

Solution 1(f)

Question 1(g)

Factorise the following by taking out the common factors:

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

Solution 1(g)

Question 1(h)

Factorise the following by taking out the common factors:

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

Solution 1(h)

Question 1(i)

Factorise the following by taking out the common factors:

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

Solution 1(i)

Question 1(j)

Factorise the following by taking out the common factors:

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

Solution 1(j)

Question 1(k)

Factorise the following by taking out the common factors:

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

Solution 1(k)

Question 2(a)

Factorise the following by grouping the terms:

15xy - 9x - 25y + 15

Solution 2(a)

15xy - 9x - 25y + 15

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

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

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

Question 2(b)

Factorise the following by grouping the terms:

15x2 + 7y - 3x - 35xy

Solution 2(b)

15x2 + 7y - 3x - 35xy

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

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

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

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

Question 2(c)

Factorise the following by grouping the terms:

9 + 3xy + x2y + 3x

Solution 2(c)

9 + 3xy + x2y + 3x

= 9 + 3xy + 3x + x2y

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

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

= (3 + xy)(3 + x)

Question 2(d)

Factorise the following by grouping the terms:

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

Solution 2(d)

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 2(e)

Factorise the following by grouping the terms:

x(x - 4)- x + 4

Solution 2(e)

x(x - 4) - x + 4

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

= (x - 4)(x - 1)

Question 2(f)

Factorise the following by grouping the terms:

2m3 - 5n2 - 5m2n + 2mn

Solution 2(f)

2m3 - 5n2 - 5m2n + 2mn

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

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

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

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

Question 2(g)

Factorise the following by grouping the terms:

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

Solution 2(g)

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

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

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

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

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

Question 2(h)

Factorise the following by grouping the terms:

9x3 + 6x2y2 - 4y3 - 6xy

Solution 2(h)

9x3 + 6x2y2 - 4y3 - 6xy

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

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

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

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

Question 2(i)

Factorise the following by grouping the terms:

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

Solution 2(i)

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 2(j)

Factorise the following by grouping the terms:

8x3 - 24x2y + 54xy2 -162y3

Solution 2(j)

8x3 - 24x2y + 54xy2 - 162y3

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

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

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

Question 2(k)

Factorise the following by grouping the terms:

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

Solution 2(k)

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 2(l)

Factorise the following by grouping the terms:

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

Solution 2(l)

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 2(m)

Factorise the following by grouping the terms:

p2x2 + (px2 + 1)x + p

Solution 2(m)

p2x2 + (px2 + 1)x + p

= p2x2 + px3 + x + p

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

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

= (p + x)(px2 + 1)

Question 2(n)

Factorise the following by grouping the terms:

x2 - (p + q)x + pq

Solution 2(n)

x2 - (p + q)x + pq

= x2 - px - qx + pq

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

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

= (x - p)(x - q)

Question 2(o)

Factorise the following by grouping the terms:

Solution 2(o)

Question 2(p)

Factorise the following by grouping the terms:

x + y + m(x + y)

Solution 2(p)

x + y + m(x + y)

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

= (x + y)(1 + m)

Question 2(q)

Factorise the following by grouping the terms:

Solution 2(q)

Question 2(r)

Factorise the following by grouping the terms:

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

Solution 2(r)

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 Exercise Ex. 5.2

Question 1(a)

Factorise the following by splitting the middle term:

x2 + 6x + 8

Solution 1(a)

x2 + 6x + 8

= x2 + 4x + 2x + 8

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

= (x + 4)(x + 2)

Question 1(b)

Factorise the following by splitting the middle term:

x2 - 11x + 24

Solution 1(b)

x2 - 11x + 24

= x2 - 8x - 3x + 24

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

= (x - 8)(x - 3)

Question 1(c)

Factorise the following by splitting the middle term:

x2 + 5x - 6

Solution 1(c)

x2 + 5x - 6

= x2 + 6x - x - 6

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

= (x + 6)(x - 1)

Question 1(d)

Factorise the following by splitting the middle term:

p2- 12p - 64

Solution 1(d)

p2 - 12p - 64

= p2 - 16p + 4p - 64

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

= (p - 16)(p + 4)

Question 1(e)

Factorise the following by splitting the middle term:

y2 - 2y - 24

Solution 1(e)

y2 - 2y - 24

= y2 - 6y + 4y - 24

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

=(y - 6)(y + 4)

Question 1(f)

Factorise the following by splitting the middle term:

3x2 + 19x - 14

Solution 1(f)

3x2 + 19x - 14

= 3x2 + 21x - 2x - 14

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

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

Question 1(g)

Factorise the following by splitting the middle term:

15a2 - 14a - 16

Solution 1(g)

15a2 - 14a - 16

= 15a2 - 24a + 10a - 16

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

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

Question 1(h)

Factorise the following by splitting the middle term:

12 + x - 6x2

Solution 1(h)

12 + x - 6x2

= 12 + 9x - 8x - 6x2

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

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

Question 1(i)

Factorise the following by splitting the middle term:

7x2 + 40x - 12

Solution 1(i)

7x2 + 40x - 12

= 7x2 + 42x - 2x - 12

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

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

Question 2(a)

Factorise the following:

5x2 - 17xy + 6y2

Solution 2(a)

5x2 - 17xy + 6y2

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

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

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

Question 2(b)

Factorise the following:

9x2 - 22xy + 8y2

Solution 2(b)

9x2 - 22xy + 8y2

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

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

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

Question 2(c)

Factorise the following:

2x3 + 5x2y - 12xy2

Solution 2(c)

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 2(d)

Factorise the following:

x2y2 + 15xy - 16

Solution 2(d)

x2y2 + 15xy - 16

= x2y2 + 16xy - xy - 16

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

= (xy + 16)(xy - 1)

Question 2(e)

Factorise the following:

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

Solution 2(e)

(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 2(f)

Factorise the following:

y2 + 3y + 2 + by + 2b

Solution 2(f)

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 2(g)

Factorise the following:

x3y3 - 8x2y2 + 15xy

Solution 2(g)

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 2(h)

Factorise the following:

Solution 2(h)

Question 2(i)

Factorise the following:

Solution 2(i)

Question 3(a)

Factorise the following:

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

Solution 3(a)

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 3(b)

Factorise the following:

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

Solution 3(b)

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 3(c)

Factorise the following:

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

Solution 3(c)

(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 3(d)

Factorise the following:

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

Solution 3(d)

(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 3(e)

Factorise the following:

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

Solution 3(e)

(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 3(f)

Factorise the following:

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

Solution 3(f)

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 3(g)

Factorise the following:

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

Solution 3(g)

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 3(h)

Factorise the following:

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

Solution 3(h)

(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 4(a)

Factorise the following:

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

Solution 4(a)

(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 4(b)

Factorise the following:

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

Solution 4(b)

(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 4(c)

Factorise the following:

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

Solution 4(c)

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 4(d)

Factorise the following:

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

Solution 4(d)

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 4(e)

Factorise the following:

Solution 4(e)

Question 4(f)

Factorise the following:

p4 + 23p2q2 + 90q4

Solution 4(f)

P4 + 23p2q2 + 90q4

= p4 + 18p2q2 + 5p2q2 + 90q4

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

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

Question 4(g)

Factorise the following:

2a3 + 5a2b - 12ab2

Solution 4(g)

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 Exercise Ex. 5.3

Question 1(a)

Factorise the following by the difference of two squares:

x2 - 16

Solution 1(a)

x2 - 16

= x2 - 42

= (x - 4)(x + 4)

Question 1(b)

Factorise the following by the difference of two squares:

64x2 - 121y2

Solution 1(b)

64x2 - 121y2

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

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

Question 1(c)

Factorise the following by the difference of two squares:

441 - 81y2

Solution 1(c)

441 - 81y2

= (21)2 - (9y)2

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

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

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

Question 1(d)

Factorise the following by the difference of two squares:

x6 - 196

Solution 1(d)

x6 - 196

= (x3)2 - (14)2

= (x3 - 14)(x3 + 14)

Question 1(e)

Factorise the following by the difference of two squares:

625 - b2

Solution 1(e)

625 - b2

= (25)2 - (b)2

= (25 - b)(25 + b)

Question 1(f)

Factorise the following by the difference of two squares:

Solution 1(f)

Question 1(g)

Factorise the following by the difference of two squares:

8xy2 - 18x3

Solution 1(g)

8xy2 - 18x3

= 2x(4y2 - 9x2)

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

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

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

Question 1(h)

Factorise the following by the difference of two squares:

16a4 - 81b4

Solution 1(h)

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 1(i)

Factorise the following by the difference of two squares:

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

Solution 1(i)

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 1(j)

Factorise the following by the difference of two squares:

(x + y)2 -1

Solution 1(j)

(x + y)2 - 1

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

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

Question 1(k)

Factorise the following by the difference of two squares:

x2 + y2 - z2 - 2xy

Solution 1(k)

x2 + y2 - z2 - 2xy

= x2 + y2 - 2xy - z2

= (x2 + y2 - 2xy) - z2

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

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

Question 1(l)

Factorise the following by the difference of two squares:

(x - 2y)2 -z2

Solution 1(l)

(x - 2y)2 - z2

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

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

Question 2(a)

Factorise the following:

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

Solution 2(a)

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 2(b)

Factorise the following:

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

Solution 2(b)

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 2(c)

Factorise the following:

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

Solution 2(c)

(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 2(d)

Factorise the following:

b2 - 2bc + c2 - a2

Solution 2(d)

b2 - 2bc + c2 - a2

= (b2 - 2bc + c2) - a2

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

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

Question 2(e)

Factorise the following:

Solution 2(e)

Question 2(f)

Factorise the following:

(x2 + y2 - z2)2 - 4x2y2

Solution 2(f)

(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 2(g)

Factorise the following:

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

Solution 2(g)

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 2(h)

Factorise the following:

4xy - x2 - 4y2 + z2

Solution 2(h)

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 2(i)

Factorise the following:

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

Solution 2(i)

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 2(j)

Factorise the following:

(x + y)3 - x - y

Solution 2(j)

(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 2(k)

Factorise the following:

y4 + y2 + 1

Solution 2(k)

y4 + y2 + 1

= y4 + 2y2 + 1 - y2

= (y2 + 1)2 - y2

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

Question 2(l)

Factorise the following:

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

Solution 2(l)

(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 3(a)

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

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

Solution 3(a)

Question 3(b)

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

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

Solution 3(b)

Question 3(c)

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

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

Solution 3(c)

Question 4(a)

Factorise:

Solution 4(a)

Question 4(b)

Factorise:

Solution 4(b)

Question 4(c)

Factorise:

x4 + y4 - 6x2y2

Solution 4(c)

Question 4(d)

Factorise:

4x4 + 25y4 + 19x2y2

Solution 4(d)

Question 4(e)

Factorise:

Solution 4(e)

Question 4(f)

Factorise:

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

Solution 4(f)