Class 8 SELINA Solutions Maths Chapter 13: Factorisation
Factorisation Exercise EX. 13(A)
Solution 1(i)
Correct option: (d) –7(x2 + 2y)
–7x2 – 14y = –7(x2 + 2y)
Solution 1(ii)
Correct option: (c) (x – y)(a – bx + by)
a(x – y) – b(x – y)2 = (x – y)[a – b(x – y)]
= (x – y)(a – bx + by)
Solution 1(iii)
Correct option: (a) (a + b)(a + c)
a2 + bc + ab + ac = a2 + ac + bc + ab
= a(a + c) + b(c + a)
= (a + b)(a + c)
Solution 1(iv)
Correct option: (c) (1 – 2x)(1 – 2x2)
1 – 2x – 2x2 + 4x3 = (1 – 2x) – (2x2 – 4x3)
= (1 – 2x) – 2x2(1 – 2x)
= (1 – 2x)(1 – 2x2)
Solution 1(v)
Correct option: (b) (x – y)(a – bx + by)
a(x – y) – b(y – x)2 = a(x – y) – b(x – y)2
= (x – y)[a – b(x – y)]
= (x – y)(a – bx + by)
**Answer in the book is given as option (a), which is not correct.**
Solution 2
17a6b8 – 34a4b6 + 51a2b4
= 17a2b4(a4b4 – 2a2b2 + 3)
Solution 3
3x5y – 27x4y2 + 12x3y3
= 3x3y(x2 – 9xy + 4y2)
Solution 4
x2(a – b) – y2(a – b) + z2(a – b)
= (a – b)(x2 – y2 + z2)
Solution 5
(x + y)(a + b) + (x – y)(a + b)
= (a + b)(x + y + x – y)
= (a + b)2x
= 2x(a + b)
Solution 6
2b(2a + b) – 3c(2a + b)
= (2a + b)(2b – 3c)
Solution 7
12abc – 6a2b2c2 + 3a3b3c3
= 3abc(4 – 2abc + a2b2c2)
Solution 8
4x(3x – 2y) – 2y(3x – 2y)
= (3x – 2y)(4x – 2y)
= (3x – 2y)2(2x – y)
= 2(3x – 2y)(2x – y)
Solution 9
(a + 2b)(3a + b) – (a + b)(a + 2b) + (a + 2b)2
= (a + 2b)[(3a + b) – (a + b) + (a + 2b)]
= (a + 2b)(3a + b – a – b + a + 2b)
= (a + 2b)(3a + 2b)
Solution 10
6xy(a2 + b2) + 8yz(a2 + b2) – 10xz(a2 + b2)
= 2(a2 + b2)(3xy + 4yz – 5xz)
Solution 11
xy – ay – ax + a2 + bx – ab
= y(x – a) – a(x – a) + b(x – a)
= (x – a)(y – a + b)
Solution 12
3x5 – 6x4 – 2x3 + 4x2 + x – 2
= (3x5 – 6x4) – (2x3 – 4x2) + (x – 2)
= 3x4(x – 2) – 2x2(x – 2) + (x – 2)
= (x – 2)(3x4 – 2x2 + 1)
Solution 13
–x2y – x + 3xy + 3
= –x(xy + 1) + 3(xy + 1)
= (xy + 1)(3 – x)
Solution 14
6a2 – 3a2b – bc2 + 2c2
= 3a2(2 – b) + c2(–b + 2)
= 3a2(2 – b) + c2(2 – b)
= (3a2 + c2)(2 – b)
Solution 15
3a2b – 12a2 – 9b + 36
= 3a2(b – 4) – 9(b – 4)
= (3a2 – 9)(b – 4)
= 3(a2 – 3)(b – 4)
Solution 16
x2 – (a – 3)x – 3a
= x2 – ax + 3x – 3a
= x(x – a) + 3(x – a)
= (x – a)(x + 3)
Solution 17
ab2 – (a – c)b – c
= ab2 – ab + bc – c
= ab(b – 1) + c(b – 1)
= (b – 1)(ab + c)
Solution 18
(a2 – b2)c + (b2 – c2)a
= a2c – b2c + b2a – c2a
= (a2c – c2a) – (b2c – b2a)
= ac(a – c) – b2(c – a)
= ac(a – c) + b2(a – c)
= (a – c)(ac + b2)
Solution 19
a3 – a2 – ab + a + b – 1
= (a3 – a2) – (ab – b) + (a – 1)
= a2(a – 1) – b(a – 1) + (a – 1)
= (a – 1)(a2 – b + 1)
Solution 20
ab(c2 + d2) – a2cd – b2cd
= abc2 + abd2 – a2cd – b2cd
= (abc2 – b2cd) + (abd2 – a2cd)
= bc(ac – bd) + ad(bd – ac)
= bc(ac – bd) – ad(ac – bd)
= (ac – bd)(bc – ad)
Solution 21
2ab2 – aby + 2cby – cy2
= (2ab2 – aby) + (2cby – cy2)
= ab(2b – y) + cy(2b – y)
= (2b – y)(ab + cy)
Solution 22
ax + 2bx + 3cx – 3a – 6b – 9c
= (ax – 3a) + (2bx – 6b) + (3cx – 9c)
= a(x – 3) + 2b(x – 3) + 3c(x – 3)
= (x – 3)(a + 2b + 3c)
Solution 23
2ab2c – 2a + 3b3c – 3b – 4b2c2 + 4c
= (2ab2c – 2a) + (3b3c – 3b) – (4b2c2 – 4c)
= 2a(b2c – 1) + 3b(b2c – 1) – 4c(b2c – 1)
= (b2c – 1)(2a + 3b – 4c)
Factorisation Exercise EX. 13(B)
Solution 1(i)
Correct option: (a) 3(x + y)(x – y)
(2x + y)2 – (2y + x)2
= 4x2 + 4xy + y2 – 4y2 – 4xy – x2
= 3x2 – 3y2
= 3(x2 – y2)
= 3(x + y)(x – y)
Solution 1(ii)
Correct option: (b) (2 – x)(12 + x)
49 – (x + 5)2
= (7)2 – (x + 5)2
= [7 – (x + 5)][7 + (x + 5)]
= (7 – x – 5)(7 + x + 5)
= (2 – x)(12 + x)
Solution 1(iii)
Correct option: (d) (a – b)(a – b + 1)
a2 – 2ab + b2 + a – b
= (a – b)2 + (a – b)
= (a – b)(a – b + 1)
Solution 1(iv)
Correct option: (d) (x – y + 1)(x – y – 1)
x2 + y2 – 2xy – 1
= (x – y)2 – (1)2
= (x – y + 1)(x – y – 1)
Solution 1(v)
Correct option: (b) (a + 1 + b – x)(a + 1 – b + x)
a2 + 2a + 1 – b2 – x2 + 2bx
= (a2 + 2a + 1) – (b2 + x2 – 2bx)
= (a + 1)2 – (b – x)2
= [(a + 1) + (b – x)][(a + 1) – (b – x)]
= (a + 1 + b – x)(a + 1 – b + x)
Solution 2
(a + 2b)2 – (a)2
= (a + 2b + a)(a + 2b – a)
= (2a + 2b)(2b)
=4b(a + b)
Solution 3
(5a – 3b)2 – 16b2
= (5a – 3b)2 – (4b)2
= (5a – 3b + 4b)(5a – 3b – 4b)
= (5a + b)(5a – 7b)
Solution 4
a4 – (a2 – 3b2)2
= (a2)2 – (a2 – 3b2)2
= [a2 – (a2 – 3b2)][a2 + (a2 – 3b2)]
= (a2 – a2 + 3b2)(a2 + a2 – 3b2)
= 3b2(2a2 – 3b2)
Solution 5
(5a – 2b)2 – (2a – b)2
= [(5a – 2b) – (2a – b)][(5a – 2b) + (2a – b)]
= (5a – 2b – 2a + b)(5a – 2b + 2a – b)
= (3a – b)(7a – 3b)
Solution 6
1 – 25(a + b)2
= (1)2 – (5)2(a + b)2
= (1)2 – [5(a + b)]2
= [1 – 5(a + b)][1 + 5(a + b)]
= (1 – 5a – 5b)(1 + 5a + 5b)
Solution 7
4(2a + b)2 – (a – b)2
= [2(2a + b)]2 – (a – b)2
= [2(2a + b) + (a – b)] [2(2a + b) – (a – b)]
= (4a + 2b + a – b)(4a + 2b – a + b)
= (5a + b)(3a + 3b)
= 3(5a + b)(a + b)
Solution 8
25(2x + y)2 – 16(x – y)2
= [5(2x + y)]2 – [4(x – y)]2
= [5(2x + y) + 4(x – y)][5(2x + y) – 4(x – y)]
= (10x + 5y + 4x – 4y)(10x + 5y – 4x + 4y)
= (14x + y)(6x + 9y)
= 2(14x + y)(2x + 3y)
Solution 9
Solution 10
(0.7)2 – (0.3)2 = (0.7 + 0.3)(0.7 – 0.3)
= (1)(0.4)
= 0.4
Solution 11
75(x + y)2 – 48(x – y)2
= (25 × 3)(x + y)2 – (16 × 3)(x – y)2
= 3(5)2(x + y)2 – 3(4)2(x – y)2
= 3[5(x + y)]2 – 3[4(x – y)]2
= 3[5(x + y) + 4(x – y)][5(x + y) – 4(x – y)]
= 3(5x + 5y + 4x – 4y)(5x + 5y – 4x + 4y)
= 3(9x + y)(x + 9y)
Solution 12
a2 + 4a + 4 – b2
= (a2 + 4a + 4) – b2
= (a + 2)2 – b2
= (a + 2 + b)(a + 2 – b)
Solution 13
a2 – b2 – 2b – 1
= a2 – (b2 + 2b + 1)
= a2 – (b + 1)2
= (a + b + 1)(a – b – 1)
Solution 14
x2 + 6x + 9 – 4y2
= (x2 + 6x + 9) – (2y)2
= (x + 3)2 – (2y)2
= (x + 3 + 2y)(x + 3 – 2y)
Factorisation Exercise EX. 13(C)
Solution 1(i)
Correct option: (a) (x – 10)(x + 1)
x2 – 9x – 10
= x2 – 10x + x – 10
= x(x – 10) + (x – 10)
= (x – 10)(x + 1)
Solution 1(ii)
Correct option: (b) (x – 21)(x – 2)
x2 – 23x + 42
= x2 – 21x – 2x + 42
= x(x – 21) – 2(x – 21)
= (x – 21)(x – 2)
Solution 1(iii)
Correct option: (b) 2x – 1
(4x2 – 4x + 1) ÷ (2x – 1)
= (2x – 1)2 ÷ (2x – 1)
= (2x – 1)
Solution 1(iv)
Correct option: (c) (x + y – 4)(x + y + 1)
(x + y)2 – 3(x + y) – 4
Taking (x + y) = a
a2 – 3a – 4
= a2 – 4a + a – 4
= a(a – 4) + (a – 4)
= (a – 4)(a + 1)
= (x + y – 4)(x + y + 1)
Solution 1(v)
Correct option: (c) (4 + x)(15 – x)
60 + 11x – x2
= 60 + 15x – 4x – x2
= 15(4 + x) – x(4 + x)
= (4 + x)(15 – x)
**Options (a) & (c) are same.**
Solution 2
a2 + 5a + 6
= a2 + 2a + 3a + 6
= a(a + 2) + 3(a + 2)
= (a + 2)(a + 3)
Solution 3
a2 – 5a + 6
= a2 – 2a – 3a + 6
= a(a – 2) – 3(a – 2)
= (a – 3)(a – 2)
Solution 4
a2 + 5a – 6
= a2 + 6a – a – 6
= a(a + 6) – (a + 6)
= (a + 6)(a – 1)
Solution 5
x2 + 5xy + 4y2
= x2 + 4xy + xy + 4y2
= x(x + 4y) + y(x + 4y)
= (x + 4y)(x + y)
Solution 6
a2 – 3a – 40
= a2 – 8a + 5a – 40
= a(a – 8) + 5(a – 8)
= (a – 8)(a + 5)
Solution 7
x2 – x – 72
= x2 – 9x + 8x – 72
= x(x – 9) + 8(x – 9)
= (x – 9)(x + 8)
Solution 8
3a2 – 5a + 2
= 3a2 – 3a – 2a + 2
= 3a(a – 1) – 2(a – 1)
= (a – 1)(3a – 2)
Solution 9
2a2 – 17ab + 26b2
= 2a2 – 13ab – 4ab + 26b2
= a(2a – 13b) – 2b(2a – 13b)
= (2a – 13b)(a – 2b)
Solution 10
2x2 + xy – 6y2
= 2x2 + 4xy – 3xy – 6y2
= 2x(x + 2y) – 3y(x + 2y)
= (x + 2y)(2x – 3y)
Solution 11
4c2 + 3c – 10
= 4c2 + 8c – 5c – 10
= 4c(c + 2) – 5(c + 2)
= (c + 2)(4c – 5)
Solution 12
14x2 + x – 3
= 14x2 + 7x – 6x – 3
= 7x(2x + 1) – 3(2x + 1)
= (2x + 1)(7x – 3)
Solution 13
6 + 7b – 3b2
= 6 + 9b – 2b – 3b2
= 3(2 + 3b) – b(2 + 3b)
= (2 + 3b)(3 – b)
Solution 14
5 + 7x – 6x2
= 5 + 10x – 3x – 6x2
= 5(1 + 2x) – 3x(1 + 2x)
= (1 + 2x)(5 – 3x)
Solution 15
4 + y – 14y2
= 4 + 8y – 7y – 14y2
= 4(1 + 2y) – 7y(1 + 2y)
= (1 + 2y)(4 – 7y)
Solution 16
5 + 3a – 14a2
= 5 + 10a – 7a – 14a2
= 5(1 + 2a) – 7a(1 + 2a)
= (1 + 2a)(5 – 7a)
Solution 17
(2a + b)2 + 5(2a + b) + 6
Taking (2a + b) = x
x2 + 5x + 6
= x2 + 3x + 2x + 6
= x(x + 3) + 2(x + 3)
= (x + 3)(x + 2)
= (2a + b + 3)(2a + b + 2)
Solution 18
1 – (2x + 3y) – 6(2x + 3y)2
Taking (2x + 3y) = a
1 – a – 6a2
= 1 – 3a + 2a – 6a2
= (1 – 3a) + 2a(1 – 3a)
= (1 – 3a)(1 + 2a)
= [1 – 3(2x + 3y)][1 + 2(2x + 3y)]
= (1 – 6x – 9y)(1 + 4x + 6y)
Solution 19
(x – 2y)2 – 12(x – 2y) + 32
Taking (x – 2y) = a
a2 – 12a + 32
= a2 – 4a – 8a + 32
= a(a – 4) – 8(a – 4)
= (a – 4)(a – 8)
= (x – 2y – 4)(x – 2y – 8)
Solution 20
8 + 6(a + b) – 5(a + b)2
Taking (a + b) = x
8 + 6x – 5x2
= 8 + 10x – 4x – 5x2
= 2(4 + 5x) – x(4 + 5x)
= (4 + 5x)(2 – x)
= [4 + 5(a + b)][2 – (a + b)]
= (4 + 5a + 5b)(2 – a – b)
Solution 21
2(x + 2y)2 – 5(x + 2y) + 2
Taking (x + 2y) = a
2a2 – 5a + 2
= 2a2 – 4a – a + 2
= 2a(a – 2) – (a – 2)
= (a – 2)(2a – 1)
= (x + 2y – 2)[2(x + 2y) – 1]
= (x + 2y – 2)(2x + 4y – 1)
Solution 22(i)
x2 + 14x + 49
= x2 + 2 × 7 × x + (7)2
= (x + 7)2
Hence, the given trinomial is a perfect square trinomial.
Solution 22(ii)
a2 – 10a + 25
= a2 – 2 × 5 × a + (5)2
= (a – 5)2
Hence, the given trinomial is a perfect square trinomial.
Solution 22(iii)
4x2 + 4x + 1
= (2x)2 + 2 × 2x × 1 + (1)2
= (2x + 1)2
Hence, the given trinomial is a perfect square trinomial.
Solution 22(iv)
9b2 + 12b + 16
= (3b)2 + 3b × 4 + (4)2
Since the given trinomial cannot be expressed in the form x2 + 2xy + y2, it is not a perfect square trinomial.
Solution 22(v)
16x2 – 16xy + y2
= (4x)2 + 2 × 4x × 2y + (y)2
Since the given trinomial cannot be expressed in the form a2 + 2ab + b2, it is not a perfect square trinomial.
Solution 22(vi)
x2 – 4x + 16
= (x)2 – x × 4 + (4)2
Since the given trinomial cannot be expressed in the form a2 – 2ab + b2, it is not a perfect square trinomial.
Factorisation Exercise EX. 13(D)
Solution 1(i)
Correct option: (b) x(x + 2)(x – 2)
x3 – 4x
= x(x2 – 4)
= x(x2 – 22)
= x(x + 2)(x – 2)
Solution 1(ii)
Correct option: (c) (x + y)(x – y)(x2 + y2 + 1)
x4 – y4 + x2 – y2
= (x4 – y4) + (x2 – y2)
= (x2 + y2)(x2 – y2) + (x2 – y2)
= (x2 – y2)(x2 + y2 + 1)
= (x + y)(x – y)(x2 + y2 + 1)
Solution 1(iii)
Correct option: (b) (x – 1)(x2 + a + 1)
x3 – x2 + ax + x – a – 1
= (x3 – x2) + (ax – a) + (x – 1)
= x2(x – 1) + a(x – 1) + (x – 1)
= (x – 1)(x2 + a + 1)
Solution 1(iv)
Correct option: (c) 2x(2x + 3)(2x – 3)
8x3 – 18x
= 2x(4x2 – 9)
= 2x[(2x)2 – 32]
= 2x(2x + 3)(2x – 3)
Solution 1(v)
Correct option: (c) (x – a)(x + b)
x2 – (a – b)x – ab
= x2 – ax + bx – ab
= x(x – a) + b(x – a)
= (x – a)(x + b)
Solution 2
8x2y – 18y3
= 2y(4x2 – 9y2)
= 2y[(2x)2 – (3y)2]
= 2y(2x + 3y)(2x – 3y)
Solution 3
25x3 – x
= x(25x2 – 1)
= x[(5x)2 – (1)2]
= x(5x + 1)(5x – 1)
Solution 4
16x4 – 81y4
= (4x2)2 – (9y2)2
= (4x2 + 9y2)(4x2 – 9y2)
= (4x2 + 9y2)[(2x)2 – (3y)2]
= (4x2 + 9y2)(2x + 3y)(2x – 3y)
Solution 5
x2 – y2 – 3x – 3y
= (x2 – y2) – (3x + 3y)
= (x + y)(x – y) – 3(x + y)
= (x + y)(x – y – 3)
Solution 6
x2 – y2 – 2x + 2y
= (x2 – y2) – (2x – 2y)
= (x + y)(x – y) – 2(x – y)
= (x – y)(x + y – 2)
Solution 7
3x2 + 15x – 72
= 3(x2 + 5x – 24)
= 3(x2 + 8x – 3x – 24)
= 3[x(x + 8) – 3(x + 8)]
= 3(x + 8)(x – 3)
Solution 8
2a2 – 8a – 64
= 2(a2 – 4a – 32)
= 2(a2 – 8a + 4a – 32
= 2[a(a – 8) + 4(a – 8)]
= 2(a – 8)(a + 4)
Solution 9
3x2y + 11xy + 6y
= y(3x2 + 11x + 6)
= y(3x2 + 9x + 2x + 6)
= y[3x(x + 3) + 2(x + 3)]
= y(x + 3)(3x + 2)
Solution 10
5ap2 + 11ap + 2a
= a(5p2 + 11p + 2)
= a(5p2 + 10p + p + 2)
= a[5p(p + 2) + (p + 2)]
= a(p + 2)(5p + 1)
Solution 11
a2 + 2ab + b2 – c2
= (a2 + 2ab + b2) – c2
= (a + b)2 – c2
= (a + b + c)(a + b – c)
Solution 12
x2 + 6xy + 9y2 + x + 3y
= (x2 + 6xy + 9y2) + (x + 3y)
= [x2 + 2 × x × 3y + (3y)2] + (x + 3y)
= (x + 3y)2 + (x + 3y)
= (x + 3y)(x + 3y + 1)
Solution 13
4a2 – 12ab + 9b2 + 4a – 6b
= (4a2 – 12ab + 9b2) + (4a – 6b)
= [(2a)2 – 2 × 2a × 3b + (3b)2] + 2(2a – 3b)
= (2a – 3b)2 + 2(2a – 3b)
= (2a – 3b)(2a – 3b + 2)
Solution 14
2a2b2 – 98b4
= 2b2(a2 – 49b2)
= 2b2[a2 – (7b)2]
= 2b2(a + 7b)(a – 7b)
Solution 15
a2 – 16b2 – 2a – 8b
= (a2 – 16b2) – (2a + 8b)
= [a2 – (4b)2] – 2(a + 4b)
= (a + 4b)(a – 4b) – 2(a + 4b)
= (a + 4b)(a – 4b – 2)
Factorisation Exercise TEST YOURSELF
Solution 1(i)
Correct option: (d) (a – b)(a – b)
(a + b)2 – 4ab
= a2 + b2 + 2ab – 4ab
= a2 + b2 – 2ab
= (a – b)2
= (a – b)(a – b)
Solution 1(ii)
Correct option: (d) (a2 + 8)(a + 2)(a – 2)
a4 + 4a2 – 32
= a4 + 8a2 – 4a2 – 32
= a2(a2 + 8) – 4(a2 + 8)
= (a2 + 8)(a2 – 4)
= (a2 + 8)(a + 2)(a – 2)
Solution 1(iii)
Correct option: (d) none of these
36 – 60y + 25y2
= 62 – 2 × 6 × 5y + (5y)2
= (6 – 5y)2
= (6 – 5y)(6 – 5y)
Solution 1(iv)
Correct option: (d) (x – 2y)(x – 2y – 3)
(x – 2y)2 – 3x + 6y
= (x – 2y)2 – (3x – 6y)
= (x – 2y)2 – 3(x – 2y)
= (x – 2y)(x – 2y – 3)
Solution 1(v)
Correct option: (d) (x – y)(ax – ay + b)
a(x – y)2 – by + bx
= a(x – y)2 + b(x – y)
= (x – y)[a(x – y) + b]
= (x – y)(ax – ay + b)
Solution 2(i)
6x3 – 8x2
= 2x2(3x – 4)
Solution 2(ii)
36x2y2 – 30x3y3 + 48x3y2
= 6x2y2(6 – 5xy + 8x)
Solution 2(iii)
8(2a + 3b)3 – 12(2a + 3b)2
= 4(2a + 3b)2[2(2a + 3b) – 3]
= 4(2a + 3b)2(4a + 6b – 3)
Solution 2(iv)
9a(x – 2y)4 – 12a(x – 2y)3
= 3a(x – 2y)3[3(x – 2y) – 4]
= 3a(x – 2y)3(3x – 6y – 4)
Solution 3(i)
a2 – ab(1 – b) – b3
= a2 – ab + ab2 – b3
= a(a – b) + b2(a – b)
= (a – b)(a + b2)
Solution 3(ii)
xy2 + (x – 1)y – 1
= xy2 + xy – y – 1
= xy(y + 1) – (y + 1)
= (y + 1)(xy – 1)
Solution 3(iii)
(ax + by)2 + (bx – ay)2
= a2x2 + 2abxy + b2y2 + b2x2 – 2abxy + a2y2
= (a2x2 + b2x2) + (b2y2 + a2y2)
= x2(a2 + b2) + y2(b2 + a2)
= (a2 + b2)(x2 + y2)
Solution 3(iv)
ab(x2 + y2) – xy(a2 + b2)
= abx2 + aby2 – a2xy – b2xy
= (abx2 – a2xy) + (aby2 – b2xy)
= ax(bx – ay) + by(ay – bx)
= ax(bx – ay) – by(bx – ay)
= (bx – ay)(ax – by)
Solution 3(v)
m – 1 – (m – 1)2 + am – a
= (m – 1) – (m – 1)2 + a(m – 1)
= (m – 1)(1 – m + 1 + a)
= (m – 1)(2 – m + a)
Solution 4(i)
25(2x – y)2 – 16(x – 2y)2
= [5(2x – y)]2 – [4(x – 2y)]2
= [5(2x – y) + 4(x – 2y)] [5(2x – y) – 4(x – 2y)]
= (10x – 5y + 4x – 8y)(10x – 5y – 4x + 8y)
= (14x – 13y)(6x + 3y)
= 3(14x – 13y)(2x + y)
Solution 4(ii)
16(5x + 4)2 – 9(3x – 2)2
= [4(5x + 4)]2 – [3(3x – 2)]2
= [4(5x + 4) + 3(3x – 2)][4(5x + 4) – 3(3x – 2)]
= (20x + 16 + 9x – 6)(20x + 16 – 9x + 6)
= (29x + 10)(11x + 22)
= 11(29x + 10)(x + 2)
Solution 4(iii)
25(x – 2y)2 – 4
= [5(x – 2y)]2 – [2]2
= [5(x – 2y) + 2][5(x – 2y) – 2]
= (5x – 10y + 2)(5x – 10y – 2)
Solution 5(i)
a2 – 23a + 42
= a2 – 21a – 2a + 42
= a(a – 21) – 2(a – 21)
= (a – 21)(a – 2)
Solution 5(ii)
a2 – 23a – 108
= a2 – 27a + 4a – 108
= a(a – 27) + 4(a – 27)
= (a – 27)(a + 4)
Solution 5(iii)
1 – 18x – 63x2
= 1 – 21x + 3x – 63x2
= (1 – 21x) + 3x(1 – 21x)
= (1 – 21x)(1 + 3x)
Solution 5(iv)
5x2 – 4xy – 12y2
= 5x2 – 10xy + 6xy – 12y2
= 5x(x – 2y) + 6y(x – 2y)
= (x – 2y)(5x + 6y)
Solution 5(v)
x(3x + 14) + 8
= 3x2 + 14x + 8
= 3x2 + 12x + 2x + 8
= 3x(x + 4) + 2(x + 4)
= (x + 4)(3x + 2)
Solution 5(vi)
5 – 4x(1 + 3x)
= 5 – 4x – 12x2
= 5 – 10x + 6x – 12x2
= 5(1 – 2x) + 6x(1 – 2x)
= (1 – 2x)(5 + 6x)
Solution 5(vii)
x2y2 – 3xy – 40
= x2y2 – 8xy + 5xy – 40
= xy(xy – 8) + 5(xy – 8)
= (xy – 8)(xy + 5)
Solution 5(viii)
(3x – 2y)2 – 5(3x – 2y) – 24
Taking 3x – 2y = a
a2 – 5a – 24
= a2 – 8a + 3a – 24
= a(a – 8) + 3(a – 8)
= (a – 8)(a + 3)
= (3x – 2y – 8)(3x – 2y + 3)
Solution 5(ix)
12(a + b)2 – (a + b) – 35
Taking (a + b) = x
12x2 – x – 35
= 12x2 + 20x – 21x – 35
= 4x(3x + 5) – 7(3x + 5)
= (3x + 5)(4x – 7)
= [3(a + b) + 5][4(a + b) – 7]
= (3a + 3b + 5)(4a + 4b – 7)
Solution 6(i)
15(5x – 4)2 – 10(5x – 4)
= 5(5x – 4)[3(5x – 4) – 2]
= 5(5x – 4)(15x – 12 – 2)
= 5(5x – 4)(15x – 14)
Solution 6(ii)
3a2x – bx + 3a2 – b
= (3a2x + 3a2) – (bx + b)
= 3a2(x + 1) – b(x + 1)
= (x + 1)(3a2 – b)
Solution 6(iii)
b(c – d)2 + a(d – c) + 3(c – d)
= b(c – d)2 – a(c – d) + 3(c – d)
= (c – d)[b(c – d) – a + 3]
= (c – d)(bc – bd – a + 3)
Solution 6(iv)
ax2 + b2y – ab2 – x2y
= (ax2 – ab2) – (x2y – b2y)
= a(x2 – b2) – y(x2 – b2)
= (x2 – b2)(a – y)
= (x + b)(x – b)(a – y)
Solution 6(v)
1 – 3x – 3y – 4(x + y)2
= 1 – 3(x + y) – 4(x + y)2
= 1 – 4(x + y) + (x + y) – 4(x + y)2
= [1 – 4(x + y)] + (x + y)[1 – 4(x + y)]
= (1 + x + y)(1 – 4x – 4y)
Solution 7(i)
2a3 – 50a
= 2a(a2 – 25)
= 2a(a2 – 52)
= 2a(a + 5)(a – 5)
Solution 7(ii)
54a2b2 – 6
= 6(9a2b2 – 1)
= 6[(3ab)2 – 12]
= 6(3ab + 1)(3ab – 1)
Solution 7(iii)
64a2b – 144b3
= 16b(4a2 – 9b2)
= 16b[(2a)2 – (3b)2]
= 16b(2a + 3b)(2a – 3b)
Solution 7(iv)
(2x – y)3 – (2x – y)
= (2x – y)[(2x – y)2 – 1]
= (2x – y)(2x – y + 1)(2x – y – 1)
Solution 7(v)
x2 – 2xy + y2 – z2
= (x2 – 2xy + y2) – z2
= (x – y)2 – z2
= (x – y + z)(x – y –z)
Solution 7(vi)
x2 – y2 – 2yz – z2
= x2 – (y2 + 2yz + z2)
= x2 – (y + z)2
= (x + y + z)(x – y – z)
Solution 7(vii)
7a5 – 567a
= 7a(a4 – 81)
= 7a[(a2)2 – 92]
= 7a(a2 + 9)(a2 – 9)
= 7a(a2 + 9)(a + 3)(a – 3)
Solution 7(viii)
Solution 8(i)
xy2 – xz2
= x(y2 – z2)
= x(y + z)(y – z)
Then, 9 × 82 – 9 × 22
= 9(8 + 2)(8 – 2)
= 9 × 10 × 6
= 540
Solution 8(ii)
xy2 – xz2
= x(y2 – z2)
= x(y + z)(y – z)
Then, 40 × 5.52 – 40 × 4.52
= 40(5.5 + 4.5)(5.5 – 4.5)
= 40 × 10 × 1
= 400
Solution 9(i)
(a – 3b)2 – 36b2
= (a – 3b)2 – (6b)2
= (a – 3b + 6b)(a – 3b – 6b)
= (a + 3b)(a – 9b)
Solution 9(ii)
25(a – 5b)2 – 4(a – 3b)2
= [5(a – 5b)]2 – [2(a – 3b)]2
= [5(a – 5b) + 2(a – 3b)][5(a – 5b) – 2(a – 3b)]
= (5a – 25b + 2a – 6b)(5a – 25b – 2a + 6b)
= (7a – 31b)(3a – 19b)
Solution 9(iii)
a2 – 0.36b2
= a2 – (0.6b)2
= (a + 0.6b)(a – 0.6b)
Solution 9(iv)
x4 – 5x2 – 36
= x4 – 9x2 + 4x2 – 36
= x2(x2 – 9) + 4(x2 – 9)
= (x2 + 4)(x2 – 9)
= (x2 + 4)(x + 3)(x – 3)
Solution 9(v)
15(2x – y)2 – 16(2x – y) – 15
Taking (2x – y) = a
15a2 – 16a – 15
= 15a2 – 25a + 9a – 15
= 5a(3a – 5) + 3(3a – 5)
= (5a + 3)(3a – 5)
= [5(2x – y) + 3][3(2x – y) – 5]
= (10x – 5y + 3)(6x – 3y – 5)
Solution 10
3012 × 300 – 3003
= 300(3012 – 3002)
= 300(301 + 300)(301 – 300)
= 300 × 601 × 1
= 180300
Solution 11(i)
(5z2 – 80) ÷ (z – 4)
Solution 11(ii)
10y(6y + 21) ÷ (2y + 7)
Solution 11(iii)
(a2 – 14a – 32) ÷ (a + 2)
Solution 11(iv)
39x3(50x2 – 98) ÷ 26x2(5x+7)
