Rs Aggarwal 2018 for Class 8 Math Chapter 1

Question 1:

Factorise:
(i) 12x + 15
(ii) 14m − 21
(iii) 9n − 12n²

Answer:

(i) $12 x + 15 = 3 (4 x + 5)$
(ii) $14 m - 21 = 7 (2 m - 3)$
(iii) $9 n - 12 n^{2} = 3 n (3 - 4 n)$

Question 2:

Factorise:
(i) 16a² − 24ab
(ii) 15ab² − 20a²b
(iii) 12x²y³ − 21x³y²

Answer:

(i) H.C.F. of $16 a^{2}$ and $24 a b$ is $8 a$ .

∴ $16 a^{2} - 24 a b = 8 a (2 a - 3 b)$

(ii) H.C.F. of $15 a b^{2}$ and $20 a^{2} b$ is $5 a b$ .

∴ $15 a b^{2} - 20 a^{2} b = 5 a b (3 b - 4 a)$

(iii) H.C.F. of $12 x^{2} y^{3}$ and $21 x^{3} y^{2}$ is $3 x^{2} y^{2}$ .

∴ $12 x^{2} y^{3} - 21 x^{3} y^{2} = 3 x^{2} y^{2} (4 y - 7 x)$

Question 3:

Factorise:
(i) 24x³ − 36x²y
(ii) 10x³ − 15x²
(iii) 36x³y − 60x²y³z

Answer:

(i) H.C.F. of $24 x^{3}$ and $36 x^{2} y$ is $6 x^{2}$ .

∴ $24 x^{3} - 36 x^{2} y = 6 x^{2} (4 x - 6 y)$

(ii) H.C.F. of $10 x^{3}$ and $15 x^{2}$ is $5 x^{3}$ .

∴ $10 x^{3} - 15 x^{2} = 5 x^{2} (2 x - 3)$

(iii) H.C.F. of $36 x^{3} y$ and $60 x^{2} y^{3} z$ is $12 x^{2} y$ .

∴ $36 x^{3} y - 60 x^{2} y^{3} z = 12 x^{2} y (3 x - 5 y^{2} z)$

Question 4:

Factorise:
(i) 9x³ − 6x² + 12x
(ii) 8x² − 72xy + 12x
(iii) 18a³b³ − 27a²b³ + 36a³b²

Answer:

(i) H.C.F. of $9 x^{3}$ , $6 x^{2}$ and $12 x$ is $3 x$ .

∴ $9 x^{3} - 6 x^{2} + 12 x = 3 x (3 x^{2} - 2 x + 4)$

(ii) H.C.F. of $8 x^{3}$ , $72 x y$ and $12 x$ is $4 x$ .

∴ $8 x^{3} - 72 x y + 12 x = 4 x (2 x^{2} - 18 y + 3)$

(iii) H.C.F. of $18 a^{3} b^{3}$ , $27 a^{2} b^{3}$ and $36 a^{3} b^{2}$ is $9 a^{2} b^{2}$ .

∴ $18 a^{3} b^{3} - 27 a^{2} b^{3} + 36 a^{3} b^{2} = 9 a^{2} b^{2} (2 a b - 3 b + 4 a)$

Question 5:

Factorise:
(i) 14x³ + 21x⁴y − 28x²y²
(ii) −5 − 10t + 20t²

Answer:

(i) H.C.F. of $14 x^{3}$ , $21 x^{4} y$ and $28 x^{2} y^{2}$ is $7 x^{2}$ .

∴ $14 x^{3} + 21 x^{4} y - 28 x^{2} y^{2} = 7 x^{2} (2 x + 3 x^{2} y - 4 y^{2})$

(ii) H.C.F. of $- 5$ , $- 10 t$ and $20 t^{2}$ is 5.

∴ $- 5 - 10 t + 20 t^{2} = 5 (- 1 - 2 t + 4 t^{2})$

Question 6:

Factorise:
(i) x(x + 3) + 5(x + 3)
(ii) 5x(x − 4) − 7(x − 4)
(iii) 2m(1 − n) + 3(1 − n)

Answer:

(i) $x (x + 3) + 5 (x + 3) = (x + 3) (x + 5)$

(ii) $5 x (x - 4) - 7 (x - 4) = (x - 4) (5 x - 7)$

(iii) $2 m (1 - n) + 3 (1 - n) = (1 - n) (2 m + 3)$

Question 7:

Factorise:
6a(a − 2b) + 5b(a − 2b)

Answer:

We have:
$6 a (a - 2 b) + 5 b (a - 2 b) = (a - 2 b) (6 a + 5 b)$

Question 8:

Factorise:
x³(2a − b) + x²(2a − b)

Answer:

We have:
$x^{3} (2 a - b) + x^{2} (2 a - b) = (2 a - b) (x^{3} + x^{2}) = x^{2} (x + 1) (2 a - b)$

Question 9:

Factorise:
9a(3a − 5b) − 12a²(3a − 5b)

Answer:

We have:
$9 a (3 a - 5 b) - 12 a^{2} (3 a - 5 b) = (3 a - 5 b) (9 a - 12 a^{2}) = 3 a (3 a - 5 b) (3 - 4 a)$

Question 10:

Factorise:
(x + 5)² − 4(x + 5)

Answer:

We have:
${(x + 5)}^{2} - 4 (x + 5) = (x + 5) \{(x + 5) - 4\}$
$= (x + 5) (x + 5 - 4) = (x + 5) (x + 1)$

∴ ${(x + 5)}^{2} - 4 (x + 5) = (x + 5) (x + 1)$

Question 11:

Factorise:
3(a − 2b)² − 5(a − 2b)

Answer:

$3 {(a - 2 b)}^{2} - 5 (a - 2 b) = (a - 2 b) \{3 (a - 2 b) - 5\}$
$= (a - 2 b) (3 a - 6 b - 5)$

∴ $3 {(a - 2 b)}^{2} - 5 (a - 2 b) = (a - 2 b) (3 a - 6 b - 5)$

Question 12:

Factorise:
2a + 6b − 3(a + 3b)²

Answer:

We have:
$2 a + 6 b - 3 {(a + 3 b)}^{2} = 2 (a + 3 b) - 3 {(a + 3 b)}^{2} = (a + 3 b) \{2 - 3 (a + 3 b)\} = (a + 3 b) (2 - 3 a - 9 b)$

∴ $2 a + 6 b - 3 {(a + 3 b)}^{2} = (a + 3 b) (2 - 3 a - 9 b)$

Question 13:

Factorise:
16(2p − 3q)² − 4(2p − 3q)

Answer:

We have:
$16 {(2 p - 3 q)}^{2} - 4 (2 p - 3 q) = (2 p - 3 q) \{16 (2 p - 3 q) - 4\} = (2 p - 3 q) (32 p - 48 q - 4)$

∴ $16 {(2 p - 3 q)}^{2} - 4 (2 p - 3 q) = (2 p - 3 q) (32 p - 48 q - 4)$

Question 14:

Factorise:
x(a − 3) + y(3 − a)

Answer:

We have:
$x (a - 3) + y (3 - a) = x (a - 3) - y (a - 3) = (a - 3) (x - y)$

∴ $x (a - 3) + y (3 - a) = (a - 3) (x - y)$

Question 15:

Factorise:
12(2x − 3y)² − 16(3y − 2x)

Answer:

We have:
$12 {(2 x - 3 y)}^{2} - 16 (3 y - 2 x) = 12 {(2 x - 3 y)}^{2} + 16 (2 x - 3 y) = (2 x - 3 y) \{12 (2 x - 3 y) + 16\} = (2 x - 3 y) (24 x - 36 y + 16)$

$∴ 12 {(2 x - 3 y)}^{2} - 16 (3 y - 2 x) = (2 x - 3 y) (24 x - 36 y + 16)$

Question 16:

Factorise:
(x + y)(2x + 5) − (x + y)(x + 3)

Answer:

We have:
$(x + y) (2 x + 5) - (x + y) (x + 3) = (x + y) \{(2 x + 5) - (x + 3)\} = (x + y) (2 x + 5 - x - 3) = (x + y) (x + 2)$

Question 17:

Factorise:
ar + br + at + bt

Answer:

By grouping the terms:
$a r + b r + a t + b t = (a r + b r) + (a t + b t) = r (a + b) + t (a + b) = (a + b) (r + t)$

∴ $a r + b r + a t + b t = (a + b) (r + t)$

Question 18:

Factorise:
x² − ax − bx + ab

Answer:

By suitably arranging the terms:
$x^{2} - a x - b x + a b = x^{2} - b x - a x + a b = (x^{2} - b x) - (a x - a b) = x (x - b) - a (x - b) = (x - b) (x - a)$

∴ $x^{2} - a x - b x + a b = (x - b) (x - a)$

Question 19:

Factorise:
ab² − bc² − ab + c²

Answer:

By suitably arranging the terms:
$a b^{2} - b c^{2} - a b + c^{2} = a b^{2} - a b - b c^{2} + c^{2} = (a b^{2} - a b) - (b c^{2} - c^{2}) = a b (b - 1) - c^{2} (b - 1) = (b - 1) (a b - c^{2})$

∴ $a b^{2} - b c^{2} - a b + c^{2} = (b - 1) (a b - c^{2})$

Question 20:

Factorise:
x² − xz + xy − yz

Answer:

By suitably arranging the terms:
$x^{2} - x z + x y - y z = x^{2} + x y - x z - y z = (x^{2} + x y) - (x z + y z) = x (x + y) - z (x + y) = (x + y) (x - z)$

∴ $x^{2} - x z + x y - y z = (x + y) (x - z)$

Question 21:

Factorise:
6ab − b² + 12ac − 2bc

Answer:

By suitably arranging the terms:
$6 a b - b^{2} + 12 a c - 2 b c = 6 a b + 12 a c - b^{2} - 2 b c = (6 a b + 12 a c) - (b^{2} + 2 b c) = 6 a (b + 2 c) - b (b + 2 c) = (b + 2 c) (6 a - b)$

∴ $6 a b - b^{2} + 12 a c - 2 b c = (b + 2 c) (6 a - b)$

Question 22:

Factorise:
(x − 2y)² + 4x − 8y

Answer:

We have:
${(x - 2 y)}^{2} + 4 x - 8 y = {(x - 2 y)}^{2} + 4 (x - 2 y) = (x - 2 y) (x - 2 y) + 4 (x - 2 y) = (x - 2 y) \{(x - 2 y) + 4\} = (x - 2 y) (x - 2 y + 4)$

∴ ${(x - 2 y)}^{2} + 4 x - 8 y = (x - 2 y) (x - 2 y + 4)$

Question 23:

Factorise:
y² − xy(1 − x) − x³

Answer:

We have:
$y^{2} - x y (1 - x) - x^{3} = y^{2} - x y + x^{2} y - x^{3} = (y^{2} - x y) + (x^{2} y - x^{3}) = y (y - x) + x^{2} (y - x) = (y - x) (y + x^{2})$

∴ $y^{2} - x y (1 - x) - x^{3} = (y - x) (y + x^{2})$

Question 24:

Factorise:
(ax + by)² + (bx − ay)²

Answer:

We have:
${(a x + b y)}^{2} + {(b x - a y)}^{2} = (a^{2} x^{2} + b^{2} y^{2} + 2 a x b y) + (b^{2} x^{2} + a^{2} y^{2} - 2 b x a y) = a^{2} x^{2} + a^{2} y^{2} + b^{2} y^{2} + b^{2} x^{2} + 2 a x b y - 2 b x a y = a^{2} (x^{2} + y^{2}) + b^{2} x^{2} + b^{2} y^{2} + 2 a x b y - 2 a x b y = a^{2} (x^{2} + y^{2}) + b^{2} (x^{2} + y^{2}) = (x^{2} + y^{2}) (a^{2} + b^{2})$

∴ ${(a x + b y)}^{2} + {(b x - a y)}^{2} = (x^{2} + y^{2}) (a^{2} + b^{2})$

Question 25:

Factorise:
ab² + (a − 1)b − 1

Answer:

We have:
$a b^{2} + (a - 1) b - 1 = a b^{2} + b a - b - 1 = (a b^{2} + b a) - (b + 1) = a b (b + 1) - 1 (b + 1) = (b + 1) (a b - 1)$

∴ $a b^{2} + (a - 1) b - 1 = (b + 1) (a b - 1)$

Question 26:

Factorise:
x³ − 3x² + x − 3

Answer:

We have:
$x^{3} - 3 x^{2} + x - 3 = (x^{3} - 3 x^{2}) + (x - 3) = x^{2} (x - 3) + 1 (x - 3) = (x - 3) (x^{2} + 1)$

∴ $x^{3} - 3 x^{2} + x - 3 = (x - 3) (x^{2} + 1)$

Question 27:

Factorise:
ab(x² + y²) − xy(a² + b²)

Answer:

We have:
$a b (x^{2} + y^{2}) - x y (a^{2} + b^{2}) = a b x^{2} + a b y^{2} - a^{2} x y - b^{2} x y = a b x^{2} - a^{2} x y + a b y^{2} - b^{2} x y = a x (b x - a y) + b y (a y - b x) = a x (b x - a y) - b y (b x - a y) = (b x - a y) (a x - b y)$

∴ $a b (x^{2} + y^{2}) - x y (a^{2} + b^{2}) = (b x - a y) (a x - b y)$

Question 28:

Factorise:
x² − x(a + 2b) + 2ab

Answer:

We have:
$x^{2} - x (a + 2 b) + 2 a b = x^{2} - a x - 2 b x + 2 a b$
$= x^{2} - 2 b x - a x + 2 a b = (x^{2} - 2 b x) - (a x - 2 a b) = x (x - 2 b) - a (x - 2 b) = (x - 2 b) (x - a)$

∴ $x^{2} - x (a + 2 b) + 2 a b = (x - 2 b) (x - a)$

Question 1:

Factorise:
x² − 36

Answer:

We have:
$x^{2} - 36 = {(x)}^{2} - {(6)}^{2} = (x + 6) (x - 6)$

∴ $x^{2} - 36 = (x + 6) (x - 6)$

Question 2:

Factorise:
4a² − 9

Answer:

We have:
$4 a^{2} - 9 = {(2 a)}^{2} - {(3)}^{2} = (2 a + 3) (2 a - 3)$

∴ $4 a^{2} - 9 = (2 a + 3) (2 a - 3)$

Question 3:

Factorise:
81 − 49x²

Answer:

We have:
$81 - 49 x^{2} = {(9)}^{2} - {(7 x)}^{2} = (9 + 7 x) (9 - 7 x)$

∴ $81 - 49 x^{2} = (9 + 7 x) (9 - 7 x)$

Question 4:

Factorise:
4x² − 9y²

Answer:

We have:
$4 x^{2} - 9 y^{2} = {(2 x)}^{2} - {(3 y)}^{2} = (2 x + 3 y) (2 x - 3 y)$

∴ $4 x^{2} - 9 y^{2} = (2 x + 3 y) (2 x - 3 y)$

Question 5:

Factorise:
16a² − 225b²

Answer:

We have:
$16 a^{2} - 225 b^{2} = {(4 a)}^{2} - {(15 b)}^{2} = (4 a + 15 b) (4 a - 15 b)$

∴ $16 a^{2} - 225 b^{2} = (4 a + 15 b) (4 a - 15 b)$

Question 6:

Factorise:
9a²b² − 25

Answer:

We have:
$9 a^{2} b^{2} - 25 = {(3 a b)}^{2} - {(5)}^{2} = (3 a b + 5) (3 a b - 5)$

∴ $9 a^{2} b^{2} - 25 = (3 a b + 5) (3 a b - 5)$

Question 7:

Factorise:
16a² − 144

Answer:

We have:
$16 a^{2} - 144 = {(4 a)}^{2} - {(12)}^{2} = (4 a + 12) (4 a - 12) = 4 (a + 3) 4 (a - 3) = 16 (a + 3) (a - 3)$

∴ $16 a^{2} - 144 = 16 (a + 3) (a - 3)$

Question 8:

Factorise:
63a² − 112b²

Answer:

We have:
$63 a^{2} - 112 b^{2} = 7 (9 a^{2} - 16 b^{2}) = 7 \{{(3 a)}^{2} - {(4 b)}^{2}\} = 7 (3 a + 4 b) (3 a - 4 b)$

∴ $63 a^{2} - 112 b^{2} = 7 (3 a + 4 b) (3 a - 4 b)$

Question 9:

Factorise:
20a² − 45b²

Answer:

We have:
$20 a^{2} - 45 b^{2} = 5 (4 a^{2} - 9 b^{2}) = 5 \{{(2 a)}^{2} - {(3 b)}^{2}\} = 5 (2 a + 3 b) (2 a - 3 b)$

∴ $20 a^{2} - 45 b^{2} = 5 (2 a + 3 b) (2 a - 3 b)$

Question 10:

Factorise:
12x² − 27

Answer:

We have:
$12 x^{2} - 27 = 3 (4 x^{2} - 9) = 3 \{{(2 x)}^{2} - {(3)}^{2}\} = 3 (2 x + 3) (2 x - 3)$

∴ $12 x^{2} - 27 = 3 (2 x + 3) (2 x - 3)$

Question 11:

Factorise:
x³ − 64x

Answer:

We have:
$x^{3} - 64 x = x (x^{2} - 64) = x \{{(x)}^{2} - {(8)}^{2}\} = x (x + 8) (x - 8)$

∴ $x^{3} - 64 x = x (x + 8) (x - 8)$

Question 12:

Factorise:
16x⁵ − 144x³

Answer:

We have:
$16 x^{5} - 144 x^{3} = 16 x^{3} (x^{2} - 9) = 16 x^{3} \{{(x)}^{2} - {(3)}^{2}\} = 16 x^{3} (x + 3) (x - 3)$

∴ $16 x^{5} - 144 x^{3} = 16 x^{3} (x + 3) (x - 3)$

Question 13:

Factorise:
3x⁵ − 48x³

Answer:

We have:
$3 x^{5} - 48 x^{3} = 3 x^{3} (x^{2} - 16) = 3 x^{3} \{{(x)}^{2} - {(4)}^{2}\} = 3 x^{3} (x + 4) (x - 4)$

∴ $3 x^{5} - 48 x^{3} = 3 x^{3} (x + 4) (x - 4)$

Question 14:

Factorise:
16p³ − 4p

Answer:

We have:
$16 p^{3} - 4 p = 4 p (4 p^{2} - 1) = 4 p \{{(2 p)}^{2} - {(1)}^{2}\} = 4 p (2 p + 1) (2 p - 1)$

∴ $16 p^{3} - 4 p = 4 p (2 p + 1) (2 p - 1)$

Question 15:

Factorise:
63a²b² − 7

Answer:

We have:
$63 a^{2} b^{2} - 7 = 7 (9 a^{2} b^{2} - 1) = 7 \{{(3 a b)}^{2} - {(1)}^{2}\} = 7 (3 a b + 1) (3 a b - 1)$

∴ $63 a^{2} b^{2} - 7 = 7 (3 a b + 1) (3 a b - 1)$

Question 16:

Factorise:
1 − (b − c)²

Answer:

We have:
$1 - {(b - c)}^{2} = {(1)}^{2} - {(b - c)}^{2} = \{1 + (b - c)\} \{1 - (b - c)\} = (1 + b - c) (1 - b + c)$

∴ $1 - {(b - c)}^{2} = (1 + b - c) (1 - b + c)$

Question 17:

Factorise:
(2a + 3b)² − 16c²

Answer:

We have:
${(2 a + 3 b)}^{2} - 16 c^{2} = {(2 a + 3 b)}^{2} - {(4 c)}^{2} = \{(2 a + 3 b) + 4 c\} \{(2 a + 3 b) - 4 c\} = (2 a + 3 b + 4 c) (2 a + 3 b - 4 c)$

∴ ${(2 a + 3 b)}^{2} - 16 c^{2} = (2 a + 3 b + 4 c) (2 a + 3 b - 4 c)$

Question 18:

Factorise:
(l + m)² − (l − m)²

Answer:

We have:
${(l + m)}^{2} - {(l - m)}^{2} = \{(l + m) + (l - m)\} \{(l + m) - (l - m)\} = (l + m + l - m) (l + m - l + m) = (2 l) (2 m)$

∴ ${(l + m)}^{2} - {(l - m)}^{2} = (2 l) (2 m)$

Question 19:

Factorise:
(2x + 5y)² − 1

Answer:

We have:
${(2 x + 5 y)}^{2} - 1 = {(2 x + 5 y)}^{2} - {(1)}^{2} = \{(2 x + 5 y) + 1\} \{(2 x + 5 y) - 1\} = (2 x + 5 y + 1) (2 x + 5 y - 1)$

∴ ${(2 x + 5 y)}^{2} - 1 = (2 x + 5 y + 1) (2 x + 5 y - 1)$

Question 20:

Factorise:
36c² − (5a + b)²

Answer:

We have:
$36 c^{2} - {(5 a + b)}^{2} = {(6 c)}^{2} - {(5 a + b)}^{2} = \{(6 c) + (5 a + b)\} \{(6 c) - (5 a + b)\} = (6 c + 5 a + b) (6 c - 5 a - b)$

∴ $36 c^{2} - {(5 a + b)}^{2} = (6 c + 5 a + b) (6 c - 5 a - b)$

Question 21:

Factorise:
(3x − 4y)² − 25z²

Answer:

We have:
${(3 x - 4 y)}^{2} - 25 z^{2} = {(3 x - 4 y)}^{2} - {(5 z)}^{2} = \{(3 x - 4 y) + 5 z\} \{(3 x - 4 y) - 5 z\} = (3 x - 4 y + 5 z) (3 x - 4 y - 5 z)$

∴ ${(3 x - 4 y)}^{2} - 25 z^{2} = (3 x - 4 y + 5 z) (3 x - 4 y - 5 z)$

Question 22:

Factorise:
x² − y² − 2y − 1

Answer:

We have:
$x^{2} - y^{2} - 2 y - 1 = x^{2} - (y^{2} + 2 y + 1)$
$= {(x)}^{2} - {(y + 1)}^{2} = \{x + (y + 1)\} \{x - (y + 1)\} = (x + y + 1) (x - y - 1)$

∴ $x^{2} - y^{2} - 2 y - 1 = (x + y + 1) (x - y - 1)$

Question 23:

Factorise:
25 − a² − b² − 2ab

Answer:

We have:
$25 - a^{2} - b^{2} - 2 a b = 25 - (a^{2} + b^{2} + 2 a b)$
$= 25 - {(a + b)}^{2} = {(5)}^{2} - {(a + b)}^{2} = \{5 + (a + b)\} \{5 - (a + b)\} = (5 + a + b) (5 - a - b)$

∴ $25 - a^{2} - b^{2} - 2 a b = (5 + a + b) (5 - a - b)$

Question 24:

Factorise:
25a² − 4b² + 28bc − 49c²

Answer:

We have:
$25 a^{2} - 4 b^{2} + 28 b c - 49 c^{2} = 25 a^{2} - (4 b^{2} - 28 b c + 49 c^{2})$
$= {(5 a)}^{2} - {(2 b - 7 c)}^{2} = \{5 a + (2 b - 7 c)\} \{5 a - (2 b - 7 c)\} = (5 a + 2 b - 7 c) (5 a - 2 b + 7 c)$

∴ $25 a^{2} - 4 b^{2} + 28 b c - 49 c^{2} = (5 a + 2 b - 7 c) (5 a - 2 b + 7 c)$

Question 25:

Factorise:
9a² − b² + 4b − 4

Answer:

We have:
$9 a^{2} - b^{2} + 4 b - 4 = 9 a^{2} - (b^{2} - 4 b + 4)$
$= {(3 a)}^{2} - {(b - 2)}^{2} = \{3 a + (b - 2)\} \{3 a - (b - 2)\} = (3 a + b - 2) (3 a - b + 2)$

∴ $9 a^{2} - b^{2} + 4 b - 4 = (3 a + b - 2) (3 a - b + 2)$

Question 26:

Factorise:
100 − (x − 5)²

Answer:

We have:
$100 - {(x - 5)}^{2} = {(10)}^{2} - {(x - 5)}^{2}$
$= \{10 + (x - 5)\} \{10 - (x - 5)\} = (10 + x - 5) (10 - x + 5) = (5 + x) (15 - x)$

∴ $100 - {(x - 5)}^{2} = (5 + x) (15 - x)$

Question 27:

Factorise:
Evaluate {(405)² − (395)²}.

Answer:

We have:
$\{{(405)}^{2} - {(395)}^{2}\} = (405 + 395) (405 - 395) = (800 \times 10) = 8000$

∴ $\{{(405)}^{2} - {(395)}^{2}\} = 8000$

Question 28:

Factorise:
Evaluate {(7.8)² − (2.2)²}.

Answer:

We have:
$\{{(7.8)}^{2} - {(2.2)}^{2}\} = (7.8 + 2.2) (7.8 - 2.2) = (10 \times 5.6) = 56$

∴ $\{{(7.8)}^{2} - {(2.2)}^{2}\} = 56$

Question 1:

Factorise:
x² + 8x + 16

Answer:

We have:
$x^{2} + 8 x + 16 = x^{2} + 2 \times x \times 4 + {(4)}^{2} = {(x + 4)}^{2}$

∴ $x^{2} + 8 x + 16 = {(x + 4)}^{2}$

Question 2:

Factorise:
x² + 14x + 49

Answer:

We have:
$x^{2} + 14 x + 49 = x^{2} + 2 \times x \times 7 + {(7)}^{2} = {(x + 7)}^{2}$

∴ $x^{2} + 14 x + 49 = {(x + 7)}^{2}$

Question 3:

Factorise:
1 + 2x + x²

Answer:

We have:
$1 + 2 x + x^{2} = x^{2} + 2 x + 1 = x^{2} + 2 \times x \times 1 + {(1)}^{2} = {(x + 1)}^{2}$

∴ $1 + 2 x + x^{2} = {(x + 1)}^{2}$

Question 4:

Factorise:
9 + 6z + z²

Answer:

We have;
$9 + 6 z + z^{2} = z^{2} + 6 z + 9 = z^{2} + 2 \times x \times 3 + {(3)}^{2} = {(z + 3)}^{2}$

∴ $9 + 6 z + z^{2} = {(z + 3)}^{2}$

Question 5:

Factorise:
x² + 6ax + 9a²

Answer:

We have:
$x^{2} + 6 a x + 9 a^{2} = x^{2} + 2 \times x \times 3 a + {(3 a)}^{2} = {(x + 3 a)}^{2}$

∴ $x^{2} + 6 a x + 9 a^{2} = {(x + 3 a)}^{2}$

Question 6:

Factorise:
4y² + 20y + 25

Answer:

We have:
$4 y^{2} + 20 y + 25 = {(2 y)}^{2} + 2 \times 2 y \times 5 + {(5)}^{2} = {(2 y + 5)}^{2}$

∴ $4 y^{2} + 20 y + 25 = {(2 y + 5)}^{2}$

Question 7:

Factorise:
36a² + 36a + 9

Answer:

We have:
$36 a^{2} + 36 a + 9 = {(6 a)}^{2} + 2 \times 6 a \times 3 + {(3)}^{2} = {(6 a + 3)}^{2}$

∴ $36 a^{2} + 36 a + 9 = {(6 a + 3)}^{2}$

Question 8:

Factorise:
9m² + 24m + 16

Answer:

We have:
$9 m^{2} + 24 m + 16 = {(3 m)}^{2} + 2 \times 3 m \times 4 + {(4)}^{2}$
$= {(3 m + 4)}^{2}$

∴ $9 m^{2} + 24 m + 16 = {(3 m + 4)}^{2}$

Question 9:

Factorise:
z² + z + $\frac{1}{4}$

Answer:

We have:
$z^{2} + z + \frac{1}{4} = z^{2} + 2 \times z \times \frac{1}{2} \times {(\frac{1}{2})}^{2}$
$= {(z + \frac{1}{2})}^{2}$

∴ $z^{2} + z + \frac{1}{4} = {(z + \frac{1}{2})}^{2}$

Question 10:

Factorise:
49a² + 84ab + 36b²

Answer:

We have:
$49 a^{2} + 84 a b + 36 b^{2} = {(7 a)}^{2} + 2 \times 7 a \times 6 b + {(6 b)}^{2}$
$= {(7 a + 6 b)}^{2}$

∴ $49 a^{2} + 84 a b + 36 b^{2} = {(7 a + 6 b)}^{2}$

Question 11:

Factorise:
p² − 10p + 25

Answer:

We have:
$p^{2} - 10 p + 25 = p^{2} - 2 \times p \times 5 + {(5)}^{2}$
$= {(p - 5)}^{2}$

∴ $p^{2} - 10 p + 25 = {(p - 5)}^{2}$

Question 12:

Factorise:
121a² − 88ab + 16b²

Answer:

We have:
$121 a^{2} - 88 a b + 16 b^{2} = {(11 a)}^{2} - 2 \times 11 a \times 4 b + {(4 b)}^{2}$
$= {(11 a - 4 b)}^{2}$

∴ $121 a^{2} - 88 a b + 16 b^{2} = {(11 a - 4 b)}^{2}$

Question 13:

Factorise:
1 − 6x + 9x²

Answer:

We have:
$1 - 6 x + 9 x^{2} = 9 x^{2} - 6 x + 1$
$= {(3 x)}^{2} - 2 \times 3 x \times 1 + {(1)}^{2} = {(3 x - 1)}^{2}$

∴ $1 - 6 x + 9 x^{2} = {(3 x - 1)}^{2}$

Question 14:

Factorise:
9y² − 12y + 4

Answer:

We have:
$9 y^{2} - 12 y + 4 = {(3 y)}^{2} - 2 \times 3 y \times 2 + {(2)}^{2}$
$= {(3 y - 2)}^{2}$

∴ $9 y^{2} - 12 y + 4 = {(3 y - 2)}^{2}$

Question 15:

Factorise:
16x² − 24x + 9

Answer:

We have:
$16 x^{2} - 24 x + 9 = {(4 x)}^{2} - 2 \times 4 x \times 3 + {(3)}^{2}$
$= {(4 x - 3)}^{2}$

∴ $16 x^{2} - 24 x + 9 = {(4 x - 3)}^{2}$

Question 16:

Factorise:
m² − 4mn + 4n²

Answer:

We have:
$m^{2} - 4 m n + 4 n^{2} = m^{2} - 2 \times m \times 2 n + {(2 n)}^{2}$
$= {(m - 2 n)}^{2}$

∴ $m^{2} - 4 m n + 4 n^{2} = {(m - 2 n)}^{2}$

Question 17:

Factorise:
a²b² − 6abc + 9c²

Answer:

We have:
$a^{2} b^{2} - 6 a b c + 9 c^{2} = {(a b)}^{2} - 2 \times a b \times 3 c + {(3 c)}^{2}$
$= {(a b - 3 c)}^{2}$

Question 18:

Factorise:
m⁴ + 2m²n² + n⁴

Answer:

We have:
$m^{4} + 2 m^{2} n^{2} + n^{4} = {(m^{2})}^{2} + 2 \times m^{2} \times n^{2} + {(n^{2})}^{2}$
$= {(m^{2} + n^{2})}^{2}$

∴ $m^{4} + 2 m^{2} n^{2} + n^{4} = {(m^{2} + n^{2})}^{2}$

Question 19:

Factorise:
(l + m)² − 4lm

Answer:

We have:
${(l + m)}^{2} - 4 l m = (l^{2} + m^{2} + 2 l m) - 4 l m$
$= l^{2} + m^{2} + 2 l m - 4 l m = l^{2} + m^{2} - 2 l m = {(l)}^{2} + {(m)}^{2} - 2 \times l \times m = {(l - m)}^{2}$

∴ ${(l + m)}^{2} - 4 l m = {(l - m)}^{2}$

Question 1:

Factorise:
x² + 5x + 6

Answer:

$T h e g i v e n e x p r e s s i o n i s x^{2} + 5 x + 6 . Find two numbers that follow the conditions given below : S u m = 5 P r o d u c t = 6 C l e a r l y, t h e n u m b e r s a r e 3 a n d 2 . x^{2} + 5 x + 6 = x^{2} + 3 x + 2 x + 6 = x (x + 3) + 2 (x + 3) = (x + 3) (x + 2)$

Question 2:

Factorise:
y² + 10y + 24

Answer:

$T h e g i v e n e x p r e s s i o n i s y^{2} + 10 y + 24 . Find two numbers that follow the conditions given below : S u m = 10 P r o d u c t = 24 C l e a r l y, t h e n u m b e r s a r e 6 a n d 4 . y^{2} + 10 y + 24 = y^{2} + 6 y + 4 y + 24 = y (y + 6) + 4 (y + 6) = (y + 6) (y + 4)$

Question 3:

Factorise:
z² + 12z + 27

Answer:

$T h e g i v e n e x p r e s s i o n i s z^{2} + 12 z + 27 . Find two numbers that follow the conditions given below : S u m = 12 P r o d u c t = 27 C l e a r l y, t h e n u m b e r s a r e 9 a n d 3 . z^{2} + 12 z + 27 = z^{2} + 9 z + 3 z + 27 = z (z + 9) + 3 (z + 9) = (z + 9) (z + 3)$

Question 4:

Factorise:
p² + 6p + 8

Answer:

$T h e g i v e n e x p r e s s i o n i s p^{2} + 6 p + 8 . Find two numbers that follow the conditions given below : S u m = 6 P r o d u c t = 8 C l e a r l y, t h e n u m b e r s a r e 4 a n d 2 . p^{2} + 6 p + 8 = p^{2} + 4 p + 2 p + 8 = p (p + 4) + 2 (p + 4) = (p + 4) (p + 2)$

Question 5:

Factorise:
x² + 15x + 56

Answer:

$T h e g i v e n e x p r e s s i o n i s x^{2} + 15 x + 56 . Find two numbers that follow the conditions given below : S u m = 15 P r o d u c t = 56 C l e a r l y, t h e n u m b e r s a r e 8 a n d 7 . x^{2} + 15 x + 56 = x^{2} + 8 x + 7 x + 56 = x (x + 8) + 7 (x + 8) = (x + 8) (x + 7)$

Question 6:

Factorise:
y² + 19y + 60

Answer:

$T h e g i v e n e x p r e s s i o n i s y^{2} + 19 y + 60 . Find two numbers that follow the conditions given below : S u m = 19 P r o d u c t = 60 C l e a r l y, t h e n u m b e r s a r e 15 a n d 4 . y^{2} + 19 y + 60 = y^{2} + 15 y + 4 y + 60 = y (y + 15) + 4 (y + 15) = (y + 15) (y + 4)$

Question 7:

Factorise:
x² + 13x + 40

Answer:

$T h e g i v e n e x p r e s s i o n i s x^{2} + 13 x + 40 . Find two numbers that follow the conditions given below : S u m = 13 P r o d u c t = 40 C l e a r l y, t h e n u m b e r s a r e 8 a n d 5 . x^{2} + 13 x + 40 = x^{2} + 8 x + 5 x + 40 = x (x + 8) + 5 (x + 8) = (x + 8) (x + 5)$

Question 8:

Factorise:
q² − 10q + 21

Answer:

$T h e g i v e n e x p r e s s i o n i s q^{2} - 10 q + 21 . Find two numbers that follow the conditions given below : S u m = - 10 P r o d u c t = 21 C l e a r l y, t h e n u m b e r s a r e - 7 a n d - 3 . q^{2} - 10 q + 21 = q^{2} - 7 q - 3 q + 21 = q (q - 7) - 3 (q - 7) = (q - 7) (q - 3)$

Question 9:

Factorise:
p² + 6p − 16

Answer:

$T h e g i v e n e x p r e s s i o n i s p^{2} + 6 p - 16 . Find two numbers that follow the conditions given below : S u m = 6 P r o d u c t = - 16 C l e a r l y, t h e n u m b e r s a r e 8 a n d - 2 . p^{2} + 6 p - 16 = p^{2} + 8 p - 2 p - 16 = p (p + 8) - 2 (p + 8) = (p + 8) (p - 2)$

Question 10:

Factorise:
x² − 10x + 24

Answer:

$T h e g i v e n e x p r e s s i o n i s x^{2} - 10 x + 24 . Find two numbers that follow the conditions given below : S u m = - 10 P r o d u c t = 24 C l e a r l y, t h e n u m b e r s a r e - 6 a n d - 4 . x^{2} - 10 x + 24 = x^{2} - 6 x - 4 x + 24 = x (x - 6) - 4 (x - 6) = (x - 6) (x - 4)$

Question 11:

Factorise:
x² − 23x + 42

Answer:

$T h e g i v e n e x p r e s s i o n i s x^{2} - 23 x + 42 . Find two numbers that follow the conditions given below : S u m = - 23 P r o d u c t = 42 C l e a r l y, t h e n u m b e r s a r e - 21 a n d - 2 . x^{2} - 23 x + 42 = x^{2} - 21 x - 2 x + 42 = x (x - 21) - 2 (x - 21) = (x - 21) (x - 2)$

Question 12:

Factorise:
x² − 17x + 16

Answer:

$T h e g i v e n e x p r e s s i o n i s x^{2} - 17 x + 16 . Find two numbers that follow the conditions given below : S u m = - 17 P r o d u c t = 16 C l e a r l y, t h e n u m b e r s a r e - 16 a n d - 1 . x^{2} - 17 x + 16 = x^{2} - 16 x - x + 16 = x (x - 16) - 1 (x - 16) = (x - 16) (x - 1)$

Question 13:

Factorise:
y² − 21y + 90

Answer:

$T h e g i v e n e x p r e s s i o n i s y^{2} - 21 y + 90 . Find two numbers that follow the conditions given below : S u m = - 21 P r o d u c t = 90 C l e a r l y, t h e n u m b e r s a r e - 15 a n d - 6 . y^{2} - 21 y + 90 = y^{2} - 15 y - 6 y + 90 = y (y - 15) - 6 (y - 15) = (y - 15) (y - 6)$

Question 14:

Factorise:
x² − 22x + 117

Answer:

$T h e g i v e n e x p r e s s i o n i s x^{2} - 22 x + 117 . Find two numbers that follow the conditions given below : S u m = - 22 P r o d u c t = 117 C l e a r l y, t h e n u m b e r s a r e - 13 a n d - 9 . x^{2} - 22 x + 117 = x^{2} - 13 x - 9 x + 117 = x (x - 13) - 9 (x - 13) = (x - 13) (x - 9)$

Question 15:

Factorise:
x² − 9x + 20

Answer:

$T h e g i v e n e x p r e s s i o n i s x^{2} - 9 x + 20 . Find two numbers that follow the conditions given below : S u m = - 9 P r o d u c t = 20 C l e a r l y, t h e n u m b e r s a r e - 5 a n d - 4 . x^{2} - 9 x + 20 = x^{2} - 5 x - 4 x + 20 = x (x - 5) - 4 (x - 5) = (x - 5) (x - 4)$

Question 16:

Factorise:
x² + x − 132

Answer:

$T h e g i v e n e x p r e s s i o n i s x^{2} + x - 132 . Find two numbers that follow the conditions given below : S u m = 1 a n d p P r o d u c t = - 132 C l e a r l y, t h e n u m b e r s a r e 12 a n d - 11 . x^{2} + x - 132 = x^{2} + 12 x - 11 x - 132 = x (x + 12) - 11 (x + 12) = (x + 12) (x - 11)$

Question 17:

Factorise:
x² + 5x − 104

Answer:

$T h e g i v e n e x p r e s s i o n i s x^{2} + 5 x - 104 . Find two numbers that follow the conditions given below : S u m = 5 P r o d u c t = - 104 C l e a r l y, t h e n u m b e r s a r e 13 a n d - 8 . x^{2} + 5 x - 104 = x^{2} + 13 x - 8 x - 104 = x (x + 13) - 8 (x + 13) = (x + 13) (x - 8)$

Question 18:

Factorise:
y² + 7y − 144

Answer:

$T h e g i v e n e x p r e s s i o n i s y^{2} + 7 y - 144 . Find two numbers that follow the conditions given below : S u m = 7 P r o d u c t = - 144 C l e a r l y, t h e n u m b e r s a r e 16 a n d - 9 . y^{2} + 7 y - 144 = y^{2} + 16 y - 9 y - 144 = y (y + 16) - 9 (y + 16) = (y + 16) (y - 9)$

Question 19:

Factorise:
z² + 19z − 150

Answer:

$T h e g i v e n e x p r e s s i o n i s z^{2} + 19 z - 150 . Find two numbers that follow the conditions given below : S u m = 19 P r o d u c t = - 150 C l e a r l y, t h e n u m b e r s a r e 25 a n d - 6 . z^{2} + 19 z - 150 = z^{2} + 25 z - 6 z - 150 = z (z + 25) - 6 (z + 25) = (z + 25) (z - 6)$

Question 20:

Factorise:
y² + y − 72

Answer:

$T h e g i v e n e x p r e s s i o n i s y^{2} + y - 72 . Find two numbers that follow the conditions given below : S u m = 1 P r o d u c t = - 72 C l e a r l y, t h e n u m b e r s a r e 9 a n d - 8 . y^{2} + y - 72 = y^{2} + 9 y - 8 y - 72 = y (y + 9) - 8 (y + 9) = (y + 9) (y - 8)$

Question 21:

Factorise:
a² + 6a − 91

Answer:

$T h e g i v e n e x p r e s s i o n i s a^{2} + 6 a - 91 . Find two numbers that follow the conditions given below : S u m = 6 P r o d u c t = - 91 C l e a r l y, t h e n u m b e r s a r e 13 a n d - 7 . a^{2} + 6 a - 91 = a^{2} + 13 a - 7 a - 91 = a (a + 13) - 7 (a + 13) = (a + 13) (a - 7)$

Question 22:

Factorise:
p² − 4p − 77

Answer:

$T h e g i v e n e x p r e s s i o n i s p^{2} - 4 p - 77 . Find two numbers that follow the conditions given below : S u m = - 4 P r o d u c t = - 77 C l e a r l y, t h e n u m b e r s a r e - 11 a n d 7 . p^{2} - 4 p - 77 = p^{2} - 11 p + 7 p - 77 = p (p - 11) + 7 (p - 11) = (p - 11) (p + 7)$

Question 23:

Factorise:
x² − 7x − 30

Answer:

$T h e g i v e n e x p r e s s i o n i s x^{2} - 7 x - 30 . Find two numbers that follow the conditions given below : S u m = - 7 P r o d u c t = - 30 C l e a r l y, t h e n u m b e r s a r e - 10 a n d 3 . x^{2} - 7 x - 30 = x^{2} - 10 x + 3 x - 30 = x (x - 10) + 3 (x - 10) = (x - 10) (x + 3)$

Question 24:

Factorise:
x² − 11x − 42

Answer:

$T h e g i v e n e x p r e s s i o n i s x^{2} - 11 x - 42 . Find two numbers that follow the conditions given below : S u m = - 11 P r o d u c t = - 42 C l e a r l y, t h e n u m b e r s a r e - 14 a n d 3 . x^{2} - 11 x - 42 = x^{2} - 14 x + 3 x - 42 = x (x - 14) + 3 (x - 14) = (x - 14) (x + 3)$

Question 25:

Factorise:
x² − 5x − 24

Answer:

$T h e g i v e n e x p r e s s i o n i s x^{2} - 5 x - 24 . Find two numbers that follow the conditions given below : S u m = - 5 P r o d u c t = - 24 C l e a r l y, t h e n u m b e r s a r e - 8 a n d 3 . x^{2} - 5 x - 24 = x^{2} - 8 x + 3 x - 24 = x (x - 8) + 3 (x - 8) = (x - 8) (x + 3)$

Question 26:

Factorise:
y² − 6y − 135

Answer:

$T h e g i v e n e x p r e s s i o n i s y^{2} - 6 y - 135 . Find two numbers that follow the conditions given below : S u m = - 6 P r o d u c t = - 135 C l e a r l y, t h e n u m b e r s a r e - 15 a n d 9 . y^{2} - 6 y - 135 = y^{2} - 15 y + 9 y - 135 = y (y - 15) + 9 (y - 15) = (y - 15) (y + 9)$

Question 27:

Factorise:
z² − 12z − 45

Answer:

$T h e g i v e n e x p r e s s i o n i s z^{2} - 12 z - 45 . Find two numbers that follow the conditions given below : S u m = - 12 P r o d u c t = - 45 C l e a r l y, t h e n u m b e r s a r e - 15 a n d 3 . z^{2} - 12 z - 45 = z^{2} - 15 z + 3 z - 45 = z (z - 15) + 3 (z - 15) = (z - 15) (z + 3)^{}$

Question 28:

Factorise:
x² − 4x − 12

Answer:

$T h e g i v e n e x p r e s s i o n i s x^{2} - 4 x - 12 . Find two numbers that follow the conditions given below : S u m = - 4 P r o d u c t = - 12 C l e a r l y, t h e n u m b e r s a r e - 6 a n d 2 . x^{2} - 4 x - 12 = x^{2} - 6 x + 2 x - 12 = x (x - 6) + 2 (x - 6) = (x - 6) (x + 2)^{}$

Question 29:

Factorise:
3x² + 10x + 8

Answer:

$T h e g i v e n e x p r e s s i o n i s 3 x^{2} + 10 x + 8 . Find two numbers that follow the conditions given below : S u m = 10 P r o d u c t = 3 \times 8 = 24 C l e a r l y, t h e n u m b e r s a r e 6 a n d 4 . 3 x^{2} + 10 x + 8 = 3 x^{2} + 10 x + 8 = 3 x^{2} + 6 x + 4 x + 8 = 3 x (x + 2) + 4 (x + 2) = (x + 2) (3 x + 4)^{}$

Question 30:

Factorise:
3y² + 14y + 8

Answer:

$T h e g i v e n e x p r e s s i o n i s 3 y^{2} + 14 y + 8 Find two numbers that follow the conditions given below : S u m = 14 P r o d u c t = 24 C l e a r l y, t h e n u m b e r s a r e 12 a n d 2 . 3 y^{2} + 14 y + 8 = 3 y^{2} + 12 y + 2 y + 8 = 3 y (y + 4) + 2 (y + 4) = (3 y + 2) (y + 4)^{}$

Question 31:

Factorise:
3z² − 10z + 8

Answer:

$T h e g i v e n e x p r e s s i o n i s 3 z^{2} - 10 z + 8 . Find two numbers that follow the conditions given below : S u m = - 10 P r o d u c t = 3 \times 8 = 24 C l e a r l y, t h e n u m b e r s a r e - 6 a n d - 4 . 3 z^{2} - 10 z + 8 = 3 z^{2} - 6 z - 4 z + 8 = 3 z (z - 2) - 4 (z - 2) = (3 z - 4) (z - 2)$

Question 32:

Factorise:
2x² + x − 45

Answer:

$T h e g i v e n e x p r e s s i o n i s 2 x^{2} + x - 45 . Find two numbers that follow the conditions given below : S u m = 1 P r o d u c t = - 45 \times 2 = - 90 C l e a r l y, t h e n u m b e r s a r e 10 a n d - 9 . 2 x^{2} + x - 45 = 2 x^{2} + 10 x - 9 x - 45 = 2 x (x + 5) - 9 (x + 5) = (2 x - 9) (x + 5)$

Question 33:

Factorise:
6p² + 11p − 10

Answer:

$T h e g i v e n e x p r e s s i o n i s 6 p^{2} + 11 p - 10 . Find two numbers that follow the conditions given below : S u m = 11 P r o d u c t = 6 \times - 10 = - 60 C l e a r l y, t h e n u m b e r s a r e 15 a n d - 4 . 6 p^{2} + 11 p - 10 = 6 p^{2} + 15 p - 4 p - 10 = 3 p (2 p + 5) - 2 (2 p + 5) = (2 p + 5) (3 p - 2)$

Question 34:

Factorise:
2x² − 17x − 30

Answer:

$T h e g i v e n e x p r e s s i o n i s 2 x^{2} - 17 x - 30 . Find two numbers that follow the conditions given below : S u m = - 17 P r o d u c t = - 30 \times 2 = - 60 C l e a r l y, t h e n u m b e r s a r e - 20 a n d 3 . 2 x^{2} - 17 x - 30 = 2 x^{2} - 20 x + 3 x - 30 = 2 x (x - 10) + 3 (x - 10) = (2 x + 3) (x - 10)$

Question 35:

Factorise:
7y² − 19y − 6

Answer:

$T h e g i v e n e x p r e s s i o n i s 7 y^{2} - 19 y - 6 . Find two numbers that follow the conditions given below : S u m = - 19 P r o d u c t = 7 \times - 6 = - 42 C l e a r l y, t h e n u m b e r s a r e - 21 a n d 2 . 7 y^{2} - 19 y - 6 = 7 y^{2} - 21 y + 2 y - 6 = 7 y (y - 3) + 2 (y - 3) = (7 y + 2) (y - 3)$

Question 36:

Factorise:
28 − 31x − 5x²

Answer:

$T h e g i v e n e x p r e s s i o n i s 28 - 31 x - 5 x^{2} . Find two numbers that follow the conditions given below : S u m = - 31 P r o d u c t = 28 \times - 5 = - 140 C l e a r l y, t h e n u m b e r s a r e - 35 a n d 4 . 28 - 31 x - 5 x^{2} = 28 + 4 x - 35 x - 5 x^{2} = 4 (x + 7) - 5 x (7 + x) = (x + 7) (4 - 5 x)$

Question 37:

Factorise:
3 + 23z − 8z²

Answer:

$T h e g i v e n e x p r e s s i o n i s 3 + 23 z - 8 z^{2} . Find two numbers that follow the conditions given below : S u m = 23 P r o d u c t = 3 \times - 8 = - 24 C l e a r l y, t h e n u m b e r s a r e 24 a n d - 1 . 3 + 23 z - 8 z^{2} = 3 + 24 z - z - 8 z^{2} = 3 (1 + 8 z) - z (1 + 8 z) = (1 + 8 z) (3 - z)$

Question 38:

Factorise:
6x² − 5x − 6

Answer:

$T h e g i v e n e x p r e s s i o n i s 6 x^{2} - 5 x - 6 . Find two numbers that follow the conditions given below : S u m = - 5 P r o d u c t = - 6 \times 6 = - 36 C l e a r l y, t h e n u m b e r s a r e - 9 a n d 4 . 6 x^{2} - 5 x - 6 = 6 x^{2} - 9 x + 4 x - 6 = 3 x (2 x - 3) + 2 (2 x - 3) = (2 x - 3) (3 x + 2)$

Question 39:

Factorise:
3m² + 24m + 36

Answer:

$T h e g i v e n e x p r e s s i o n i s 3 m^{2} + 24 m + 36 . Find two numbers that follow the conditions given below : S u m = 24 P r o d u c t = 36 \times 3 = 108 C l e a r l y, t h e n u m b e r s a r e 18 a n d 6 . 3 m^{2} + 24 m + 36 = 3 m^{2} + 18 m + 6 m + 36 = 3 m (m + 6) + 6 (m + 6) = (3 m + 6) (m + 6) = 3 (m + 2) (m + 6)$

Question 40:

Factorise:
4n² − 8n + 3

Answer:

$T h e g i v e n e x p r e s s i o n i s 4 n^{2} - 8 n + 3 . Find two numbers that follow the conditions given below : S u m = - 8 P r o d u c t = 4 \times 3 = 12 C l e a r l y, t h e n u m b e r s a r e - 6 a n d - 2 . 4 n^{2} - 8 n + 3 = 4 n^{2} - 2 n - 6 n + 3 = 2 n (2 n - 1) - 3 (2 n - 1) = (2 n - 1) (2 n - 3)$

Question 41:

Factorise:
6x² − 17x − 3

Answer:

$T h e g i v e n e x p r e s s i o n i s 6 x^{2} - 17 x - 3 . Find two numbers that follow the conditions given below : S u m = - 17 P r o d u c t = 6 \times - 3 = - 18 C l e a r l y, t h e n u m b e r s a r e - 18 a n d 1 . 6 x^{2} - 17 x - 3 = 6 x^{2} - 18 x + x - 3 = 6 x (x - 3) + 1 (x - 3) = (6 x + 1) (x - 3)$

Question 42:

Factorise:
7x² − 19x − 6

Answer:

$T h e g i v e n e x p r e s s i o n i s 7 x^{2} - 19 x - 6 . Find two numbers that follow the conditions given below : S u m = - 19 P r o d u c t = 7 \times - 6 = - 42 C l e a r l y, t h e n u m b e r s a r e - 21 a n d 2 . 7 x^{2} - 19 x - 6 = 7 x^{2} - 21 x + 2 x - 6 = 7 x (x - 3) + 2 (x - 3) = (7 x + 2) (x - 3)$

Question 1:

Tick (✓) the correct answer:
(7a² − 63b²) = ?
(a) (7a − 9b)(9a + 7b)
(b) (7a − 9b)(7a + 9b)
(c) 9(a − 3b)(a + 3b)
(d) 7(a − 3b)(a + 3b)

Answer:

(d) 7(a − 3b)(a + 3b)

$(7 a^{2} - 63 b^{2}) = 7 (a^{2} - 9 b^{2}) = 7 (a - 3 b) (a + 3 b)$

Question 2:

Tick (✓) the correct answer:
(2x − 32x³) = ?
(a) 2(x − 4)(x + 4)
(b) 2x(1 − 2x)²
(c) 2x(1 + 2x)²
(d) 2x(1 − 4x)(1 + 4x)

Answer:

(d) 2x(1 − 4x)(1 + 4x)

$(2 x - 32 x^{3}) = 2 x (1 - 16 x^{2}) = 2 x (1 - 4 x) (1 + 4 x)$

Question 3:

Tick (✓) the correct answer:
x³ − 144x = ?
(a) x(x − 12)²
(b) x(x + 12)²
(c) x(x − 12)(x + 12)
(d) none of these

Answer:

(c) x(x − 12)(x + 12)

$x^{3} - 144 x = x (x^{2} - 144) = x (x - 12) (x + 12)$

Question 4:

Tick (✓) the correct answer:
(2 − 50x²) = ?
(a) 2(1 − 5x)²
(b) 2(1 + 5x)²
(c) (2 − 5x)(2 + 5x)
(d) 2(1 − 5x)(1 + 5x)

Answer:

(d) 2(1 − 5x)(1 + 5x)

$2 - 50 x^{2} = 2 (1 - 25 x^{2}) = 2 (1 - 5 x) (1 + 5 x)$

Question 5:

Tick (✓) the correct answer:
a² + bc + ab + ac = ?
(a) (a + b)(a + c)
(b) (a + b)(b + c)
(c) (b + c)(c + a)
(d) a(a + b + c)

Answer:

(a) (a + b)(a + c)

$a^{2} + b c + a b + a c = a^{2} + a b + b c + a c = a (a + b) + c (a + b) = (a + c) (a + b)$

Question 6:

Tick (✓) the correct answer:
pq² + q(p − 1) − 1 = ?
(a) (pq + 1)(q − 1)
(b) p(q + 1)(q − 1)
(c) q(p − 1)(q + 1)
(d) (pq − 1) (q + 1)

Answer:

(d) (pq − 1) (q + 1)

$p q^{2} + q (p - 1) - 1 = p q^{2} + q p - q - 1 = p q (q + 1) - 1 (q + 1) = (p q - 1) (q + 1)$

Question 7:

Tick (✓) the correct answer:
ab − mn + an − bm = ?
(a) (a − b)(m − n)
(b) (a − m)(b + n)
(c) (a − n)(m + b)
(d) (m − a)(n − b)

Answer:

(b) (a − m)(b + n)

$a b - m n + a n - b m = a b + a n - m n - b m = a (b + n) - m (n + b) = (a - m) (b + n)$

Question 8:

Tick (✓) the correct answer:
ab − a − b + 1 = ?
(a) (a − 1)(b − 1)
(b) (1 − a)(1 − b)
(c) (a − 1)(1 − b)
(d) (1 − a)(b − 1)

Answer:

(a) (a − 1)(b − 1)

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

Question 9:

Tick (✓) the correct answer:
x² − xz + xy − yz = ?
(a) (x − y)(x + z)
(b) (x − y)(x − z)
(c) (x + y)(x − z)
(d) (x − y)(z − x)

Answer:

(c) (x + y)(x − z)

$x^{2} - x z + x y - y z = x (x - z) + y (x - z) = (x + y) (x - z)$

Question 10:

Tick (✓) the correct answer:
(12m² − 27) = ?
(a) (2m − 3)(3m − 9)
(b) 3(2m − 9)(3m − 1)
(c) 3(2m − 3)(2m + 3)
(d) none of these

Answer:

(c) 3(2m − 3)(2m + 3)

$12 m^{2} - 27 = 3 (4 m^{2} - 9) = 3 (2 m - 3) (2 m + 3)$

Question 11:

Tick (✓) the correct answer:
x³ − x = ?
(a) x(x² − x)
(b) x(x − x²)
(c) x(1 + x)(1 − x)
(d) x(x + 1)(x − 1)

Answer:

(d) x(x + 1)(x − 1)

$x^{3} - x = x (x^{2} - 1) = x (x - 1) (x + 1)$

Question 12:

Tick (✓) the correct answer:
1 − 2ab − (a² + b²) = ?
(a) (1 + a − b)(1 + a + b)
(b) (1 + a + b)(1 − a + b)
(c) (1 + a + b)(1 − a − b)
(d) (1 + a − b)(1 − a + b)

Answer:

(c) (1 + a + b)(1 − a − b)

$1 - 2 a b - (a^{2} + b^{2}) = 1 - 2 a b - a^{2} - b^{2} = 1 - (2 a b + a^{2} + b^{2}) = 1 - (a + b)^{2} = (1 - a - b) (1 + a + b)$

Question 13:

Tick (✓) the correct answer:
x² + 6x + 8 = ?
(a) (x + 3)(x + 5)
(b) (x + 3)(x + 4)
(c) (x + 2)(x + 4)
(d) (x + 1)(x + 8)

Answer:

(c) (x + 2)(x + 4)

$x^{2} + 6 x + 8 = x^{2} + 4 x + 2 x + 8 = x (x + 4) + 2 (x + 4) = (x + 2) (x + 4)$

Question 14:

Tick (✓) the correct answer:
x² + 4x − 21 = ?
(a) (x − 7)(x + 3)
(b) (x + 7)(x − 3)
(c) (x − 7)(x − 3)
(d) (x + 7)(x + 3)

Answer:

(b) (x + 7)(x − 3)

$x^{2} + 4 x - 21 = x^{2} + 7 x - 3 x - 21 = x (x + 7) - 3 (x + 7) = (x - 3) (x + 7)$

Question 15:

Tick (✓) the correct answer:
y² + 2y − 3 = ?
(a) (y − 1)(y + 3)
(b) (y + 1)(y − 3)
(c) (y − 1)(y − 3)
(d) (y + 2)(y − 3)

Answer:

(a) (y − 1)(y + 3)

$y^{2} + 2 y - 3 = y^{2} + 3 y - y - 3 = y (y + 3) - 1 (y + 3) = (y - 1) (y + 3)$

Question 16:

Tick (✓) the correct answer:
40 + 3x − x² = ?
(a) (x + 5)(x − 8)
(b) (5 − x)(8 + x)
(c) (5 + x)(8 − x)
(d) (5 − x)(8 − x)

Answer:

(c) (5 + x)(8 − x)

$40 + 3 x - x^{2} = 40 + 8 x - 5 x - x^{2} = 8 (5 + x) - x (5 + x) = (8 - x) (x + 5)$

Question 17:

Tick (✓) the correct answer:
2x² + 5x + 3 = ?
(a) (x + 3)(2x + 1)
(b) (x + 1)(2x + 3)
(c) (2x + 5)(x − 3)
(d) none of these

Answer:

(b) (x + 1)(2x + 3)

$2 x^{2} + 5 x + 3 = 2 x^{2} + 2 x + 3 x + 3 = 2 x (x + 1) + 3 (x + 1) = (2 x + 3) (x + 1)$

Question 18:

Tick (✓) the correct answer:
6a² − 13a + 6 = ?
(a) (2a + 3)(3a − 2)
(b) (2a − 3)(3a + 2)
(c) (3a − 2)(2a − 3)
(d) (3a + 1)(2a − 3)

Answer:

(c) (3a − 2)(2a − 3)

$6 a^{2} - 13 a + 6 = 6 a^{2} - 9 a - 4 a + 6 = 3 a (2 a - 3) - 2 (2 a - 3) = (3 a - 2) (2 a - 3)$

Question 19:

Tick (✓) the correct answer:
4z² − 8z + 3 = ?
(a) (2z − 1)(2z − 3)
(b) (2z + 1)(3 − 2z)
(c) (2z + 3)(3z − 1)
(c) (z − 1)(4z − 3)

Answer:

(a) (2z − 1)(2z − 3)

$4 z^{2} - 8 z + 3 = 4 z^{2} - 6 z - 2 z + 3 = 2 z (2 z - 3) - 1 (2 z - 3) = (2 z - 1) (2 z - 3)$

Question 20:

Tick (✓) the correct answer:
3 + 23y − 8y² = ?
(a) (1 − 8y)(3 + y)
(b) (1 + 8y)(3 − y)
(c) (1 − 8y)(y − 3)
(d) (8y − 1)(y + 3)

Answer:

(b) (1 + 8y)(3 − y)

$3 + 23 y - 8 y^{2} = 3 + 24 y - y - 8 y^{2} = 3 (1 + 8 y) - y (1 + 8 y) = (3 - y) (1 + 8 y)$

Rs Aggarwal 2018 for Class 8 Math Chapter 1 - Rational Numbers

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