Class 8 MAHARASHTRA STATE TEXTBOOK BUREAU Solutions Maths Chapter 6: Factorisation of Algebraic expressions

Here, 18 is multiplication of 6 and 3 also, 9x = 6x + 3x.

Hence,

x² + 9x + 18

= x² + 6x + 3x + 18

= x (x + 6) + 3(x + 6)

= (x + 6)(x + 3)

Here, 9 is multiplication of 9 and 1 also, -10x = -9x - x.

Hence,

x² - 10x + 9

= x² - 9x - x + 9

= x(x - 9) - (x - 9)

= (x - 9)(x - 1)

Here, 144 is multiplication of 12 and 12 also, 24y = 12y + 12y.

Hence,

y² + 24y + 144

= y² + 12y + 12y + 144

= y(y + 12) + 12(y + 12)

= (y + 12)(y + 12)

5y² + 5y - 10

= 5(y² + y - 2)

Here, 2 is multiplication of 1 and 2 also, y = 2y - y.

Hence,

= 5(y² + y - 2)

= 5(y² + 2y - y - 2)

= 5[y(y + 2) - (y + 2)]

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

Here, 35 is multiplication of 7 and 5 also, -2p = -7p + 5p.

Hence,

p² - 2p - 35

= p² - 7p + 5p - 35

= p(p - 7) + 5(p - 7)

= (p - 7)(p + 5)

Here, 44 is multiplication of 11 and 4 also, -7p = -11p + 4p.

Hence,

p² - 7p - 44

= p² - 11p + 4p - 44

= p(p - 11) + 4(p - 11)

= (p - 11)(p + 4)

Here, 120 is multiplication of 15 and 8 also, -23m = -15m - 8m.

Hence,

m² - 23m + 120

= m² - 15m - 8m + 120

= m(m - 15) - 8 (m - 15)

= (m - 15)(m - 8)

Here, 100 is multiplication of 20 and 5 also, -25m = -20m - 5m.

Hence,

m² - 25m + 100

= m² - 20m - 5m + 100

= m(m - 20) - 5(m - 20)

= (m - 20)(m - 5)

Here, 15 × 3 (Coefficient of x²) = 45 is multiplication of 5 and 9 also, 14m = 9m + 5m.

Hence,

3x² + 14x + 15

= 3x² + 9x + 5x + 15

= 3x(x + 3) + 5(x + 3)

= (x + 3)(3x + 5)

Here, 45 × 2(Coefficient of x²) = 90 is multiplication of 10 and 9 also,

x = 10x - 9x

Hence,

2x² + x - 45

= 2x² + 10x - 9x - 45

= 2x(x + 5) - 9(x + 5)

= (x + 5)(2x - 9)

Here, 8 × 20(Coefficient of x²) = 160 is multiplication of 16 and 10 also,

-26x = -16x - 10x

Hence,

20x² - 26x + 8

= 20x² - 16x - 10x + 8

= 4x(5x - 4) - 2(5x - 4)

= (5x - 4)(4x - 2)

= (5x - 4)2(2x - 1)

= 2(5x - 4)(2x - 1)

Here, 3 × 44(Coefficient of x²) = 132 is multiplication of 11 and 12 also,

-x = -12x + 11x

Hence,

44x² - x - 3

= 44x² - 12x + 11x - 3

= 4x(11x - 3) + (11x - 3)

= (11x - 3)(4x + 1)

x³ + 64y³ = x³ + (4y)³

We know that a³ + b³ = (a + b)(a² - ab + b²)

Comparing x³ + (4y)³ with a³ + b³ we get

a = x and b = 4y

x³ + 64y³

= (x + 4y)[x² - 4xy + (4y)²]

= (x + 4y) (x² - 4xy + 16y²)

125p³ + q³= (5p)³ + q³

We know that a³ + b³ = (a + b)(a² - ab + b²)

Comparing (5p)³ + q³ with a³ + b³ we get

a = 5p and b = q

125p³ + q³

= (5p + q)[(5p)² - 5pq + q²]

= (5p + q) (25p² - 5pq + q²)

125k³ + 27m³ = (5k)³ + (3m)³

We know that a³ + b³ = (a + b)(a² - ab + b²)

Comparing (5k)³ + (3m)³ with a³ + b³ we get

a = 5k and b = 3m

125k³ + 27m³

= (5k + 3m)[(5k)² - 5k × 3m + (3m)²]

= (5k + 3m) (25k² - 15km + 9m²)

2l³ + 432m³ = 2(l³ + 216m³) = 2[l³ + (6m)³]

We know that a³ + b³ = (a + b)(a² - ab + b²)

Comparing (l³ + 216m³) with a³ + b³ we get

a = l and b = 6m

2l³ + 432m³

= 2[l³ + (6m)³]

= 2(l + 6m)(l² - l × 6m + (6m)²]

= 2(l + 6m)(l² - 6lm + 36m²)

24a³ + 81b³ = 3(8a³ + 27b³) = 3[(2a)³ + (3b)³]

We know that A³ + B³ = (A + B)(A² - AB + B²)

Comparing [8a³ + (3b)³] with A³ + B³ we get

A = 2a and B = 3b

24a³ + 81b³

= 3(8a³ + 27b³)

= 3[(2a)³ + (3b)³]

= 3(2a + 3b)[(2a)² - 2a × 3b + (3b)²]

= 3(2a+ 3b)(4a² - 6ab + 9b²)

=  

We know that a³ + b³ = (a + b)(a² - ab + b²)

Comparing with a³ + b³ we get

a = y and b =  

=  

=  

=  

We know that A³ + B³ = (A + B)(A² - AB + B²)

Comparing with A³ + B³ we get

A = a and B =  

=  

=  

 =  

We know that a³ + b³ = (a + b)(a² - ab + b²)

Comparingwith a³ + b³ we get

a = 1 and b =  

=  

=  

y³ - 27 = y³ - 3³

We know that a³ - b³ = (a - b)(a² + ab + b²)

Comparing y³ - 3³ with a³ - b³ we get

a = y and b = 3

y³ - 27

= (y - 3)(y² + 3y + 3²)

= (y - 3)(y² + 3y + 9)

x³ - 64y³ = x³ - (4y)³

We know that a³ - b³ = (a - b)(a² + ab + b²)

Comparing x³ - (4y)³ with a³ - b³ we get

a = x and b = 4y

x³ - 64y³

= (x - 4y)[x² + 4xy + (4y)²]

= (x - 4y)(x² + 4xy + 16y²)

27m³ - 216n³

= 27(m³ - 8n³)

=27[m³ - (2n)³]

We know that a³ - b³ = (a - b)(a² + ab + b²)

Comparing m³ - (2n)³ with a³ - b³ we get

a = m and b = 2n

27m³ - 216n³

= 27[m³ - (2n)³]

= 27(m - 2n)[m² + m × 2n + (2n)²]

= 27(m - 2n)(m² + 2mn + 4n²)

125y³ - 1

= (5y)³ - 1

We know that a³ - b³ = (a - b)(a² + ab + b²)

Comparing (5y)³ - 1 with a³ - b³ we get

a = 5y and b = 1

125y³ - 1

= (5y - 1)[(5y)² + 5y + 1]

= (5y - 1)(25y² + 5y + 1)

  =  

We know that a³ - b³ = (a - b)(a² + ab + b²)

Comparing  with a³ - b³ we get

a = 2p and b =  

=  

=  

343a³ - 512b³

= (7a)³ - (8b)³

We know that A³ - B³ = (A - B)(A² + AB + B²)

Comparing (7a)³ - (8b)³ with A³ - B³ we get

A = 7a and B = 8b

343a³ - 512b³

= (7a - 8b)[(7a)² + 7a × 8b + (8b)²]

= (7a - 8b)(49a² + 56ab + 64b²)

64x³ - 729y³

= (4x)³ - (9y)³

We know that a³ - b³ = (a - b)(a² + ab + b²)

Comparing (4x)³ - (9y)³ with a³ - b³ we get

a = 4x and b = 9y

64x³ - 729y³

= (4x - 9y)[(4x)² + 4x × 9y + (9y)²]

= (4x - 9y)(16x² + 36xy + 81y²)

 =  

We know that A³ - B³ = (A - B)(A² + AB + B²)

Comparing  with A³ - B³ we get

A = a and B =  

=  

=  

=  

we know that (a + b)³ = a³ + 3a²b + 3ab² + b³

Consider, (x + y)³ = x³ + 3x²y + 3xy² + y³

Also, (a - b)³ = a³ - 3a²b + 3ab² - b³

Consider, (x - y)³ = x³ - 3x²y + 3xy² - y³

(x + y)³ - (x - y)³

= x³ + 3x²y + 3xy² + y³ - (x³ - 3x²y + 3xy² - y³)

= x³ + 3x²y + 3xy² + y³ - x³ + 3x²y - 3xy² + y³

= 3x²y + 3x²y + y³ + y³

= 6x²y + 2y³

= 2y(3x² + y²) 

we know that (a + b)³ = a³ + 3a²b + 3ab² + b³

Consider, (3a + 5b)³ = (3a)³ + 3(3a)²(5b) + 3(3a)(5b)² + (5b)³

= 27a³ + 135a²b + 225ab² + 125b³

Also, (a - b)³ = a³ - 3a²b + 3ab² - b³

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

= 27a³ - 135a²b + 225ab² - 125b³

(3a + 5b)³ - (3a - 5b)³

= 27a³ + 135a²b + 225ab² + 125b³ - (27a³ - 135a²b + 225ab² - 125b³)

= 27a³ + 135a²b + 225ab² + 125b³ - 27a³ + 135a²b - 225ab² + 125b³

= 135a²b + 135a²b + 125b³ + 125b³

= 270a²b + 250b³

we know that (a + b)³ = a³ + 3a²b + 3ab² + b³

(a + b)³ - a ³ - b³

= a³ + 3a²b + 3ab² + b³ - a³ - b³

= 3a²b + 3ab²

we know that (a + b)³ = a³ + 3a²b + 3ab² + b³

p³ - (p + 1)³

= p³ - (p³ + 3p² + 3p + 1)

= p³ - p³ - 3p² - 3p - 1

= - 3p² - 3p - 1

we know that (a - b)³ = a³ - 3a²b + 3ab² - b³

(3xy - 2ab)³ = (3xy)³ - 3 × (3xy)² × 2ab + 3 × (3xy) × (2ab)² - (2ab)³

= 27x³y³ - 3 × 9x²y² × 2ab + 9xy × 4a²b² - 8a³b³

= 27x³y³ - 54x²y²ab + 36xya²b² - 8a³b³

Also, (a + b)³ = a³ + 3a²b + 3ab² + b³

(3xy + 2ab)³ = (3xy)³ + 3 × (3xy)² × 2ab + 3 × (3xy) × (2ab)² + (2ab)³

= 27x³y³ + 3 × 9x²y² × 2ab + 9xy × 4a²b² + 8a³b³

= 27x³y³ + 54x²y²ab + 36xya²b² + 8a³b³

(3xy - 2ab)³ - (3xy + 2ab)³

= 27x³y³ - 54x²y²ab + 36xya²b² - 8a³b³

= - (27x³y³ + 54x²y²ab + 36xya²b² + 8a³b³

= - 54x²y²ab - 54x²y²ab - 8a³b³ - 8a³b³

= -108 x²y²ab - 8a³b³

We know that m² - n² = (m + n)(m - n) and

m³ - n³ = (m - n)(m² + mn + n²)

=  

=  

Consider,

a² + 10a + 21

= a² + 7a + 3a + 21

= a(a + 7) + 3(a + 7)

= (a + 7)(a + 3)

a² + 6a - 7

= a² + 7a - a - 7

= a(a + 7) - (a + 7)

= (a + 7)(a - 1)

Consider, 

a² - 1 = (a + 1)(a - 1)

Now,

=  

= a + 1

We know that 8x³ - 27y³ = (2x - 3y)[(2x)² + (2x)(3y) + (3y)²]

= (2x - 3y)[4x² + 6xy + 9y²]

Also, 4x² - 9y² = (2x)² - (3y)²

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

Now,

=  

=  

Consider,

x² - 5x - 24 = x² - 8x + 3x - 24

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

= (x - 8)(x + 3)

Also, x² - 64 = (x + 8)(x - 8)

Now,

=  

= 1

Consider,

3x² - x - 2 = 3x² - 3x + 2x - 2

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

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

Consider,

x² - 7x + 12

= x² - 4x - 3x + 12

= x(x - 4) - 3(x - 4)

= (x - 4)(x - 3)

Consider,

3x² - 7x - 6

= 3x² - 9x + 2x - 6

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

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

=  

=  

=  

Consider,

4x² - 11x + 6

= 4x² - 8x - 3x + 6

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

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

Consider,

16x² - 9

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

Now,

=  

=  

Consider,

a³ - 27 = (a - 3)(a² + 3a + 9) 

Consider,

5a² - 16a + 3

= 5a² - 15a - a + 3

= 5a(a - 3) - (a - 3)

= (a - 3)(5a - 1)

Consider,

25a² - 1

= (5a + 1)(5a - 1)

Now,

=  

=  

= 5a + 1

Consider,

1 - 2x + x²

= 1 - x - x + x²

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

= (1 - x)(1 - x) 

Consider,

1 - x³ = (1 - x)(1 + x + x²)

Now,

=  

=