Factorise each of the following: 
(i) 8a³ + b³ + 12a²b + 6ab²                           (ii) 8a³ - b³ - 12a²b + 6ab²
(iii) 27 - 125a³ - 135a + 225a²                      (iv) 64a³ - 27b³ - 144a²b + 108ab²
(v) 27p³ - 1/216 - 9/2 p² + 1/4 p

Answer


(i) 8a³ + b³ + 12a²b + 6ab²
Using identity, (a + b)³ = a³ + b³ + 3a²b + 3ab²
8a³ + b³ + 12a²b + 6ab² 
= (2a)³ + b³ + 3(2a)²b + 3(2a)(b)²
= (2a + b)³
= (2a + b)(2a + b)(2a + b)

(ii) 8a³ - b³ - 12a²b + 6ab²
Using identity, (a - b)³ = a³ - b³ - 3a²b + 3ab²
8a³ - b³ - 12a²b + 6ab²= (2a)³ - b³ - 3(2a)²b + 3(2a)(b)²
= (2a - b)³
= (2a - b)(2a - b)(2a - b)

(iii) 27 - 125a³ - 135a + 225a²
Using identity, (a - b)³ = a³ - b³ - 3a²b + 3ab²
27 - 125a³ - 135a + 225a²= 3³ - (5a)³ - 3(3)²(5a) + 3(3)(5a)²
= (3 - 5a)³
= (3 - 5a)(3 - 5a)(3 - 5a)

(iv) 64a³ - 27b³ - 144a²b + 108ab²
Using identity, (a - b)³ = a³ - b³ - 3a²b + 3ab²
64a³ - 27b³ - 144a²b + 108ab²= (4a)³ - (3b)³ - 3(4a)²(3b) + 3(4a)(3b)²
= (4a - 3b)³
= (4a - 3b)(4a - 3b)(4a - 3b)

(v) 27p³ - 1/216 - 9/2 p² + 1/4 p 

 Using identity, (a - b)³ = a³ - b³ - 3a²b + 3ab²
 27p³ - 1/216 - 9/2 p² + 1/4 p
= (3p)³ - (1/6)³ - 3(3p)²(1/6) + 3(3p)(1/6)²
= (3p - 1/6)³
= (3p - 1/6)(3p - 1/6)(3p - 1/6)

Verify : (i) x³ + y³ = (x + y) (x² - xy + y²)             (ii) x³ - y³ = (x - y) (x² + xy + y²)

Answer

(i) x³ + y³ = (x + y) (x² - xy + y²)
We know that, 
(x + y)³ = x³ + y³ + 3xy(x + y) 
⇒ x³ + y³ = (x + y)³ - 3xy(x + y)
⇒ x³ + y³ = (x + y)[(x + y)² - 3xy]                  {Taking (x+y) common}
⇒ x³ + y³ = (x + y)[(x² + y² + 2xy) - 3xy] 
⇒ x³ + y³ = (x + y)(x² + y² - xy) 

(ii) x³ - y³ = (x - y) (x² + xy + y² )
We know that, 
(x - y)³ = x³ - y³ - 3xy(x - y) 
⇒ x³ - y³ = (x - y)³ + 3xy(x - y)
⇒ x³ + y³ = (x - y)[(x - y)² + 3xy]                     {Taking (x-y) common}
⇒ x³ + y³ = (x - y)[(x² + y² - 2xy) + 3xy] 
⇒ x³ + y³ = (x + y)(x² + y² + xy)

Factorise each of the following:
      (i) 27y³ + 125z³                     (ii) 64m³ - 343n³

Answer

(i) 27y³ + 125z³
Using identity, x³ + y³ = (x + y) (x² - xy + y²)
27y³ + 125z³ = (3y)³ + (5z)³
= (3y + 5z) {(3y)² - (3y)(5z) + (5z)²}
= (3y + 5z) (9y² - 15yz + 25z)² 

(ii) 64m³ - 343n³ 
Using identity, x³ - y³ = (x - y) (x² + xy + y² ) 
64m³ - 343n³ = (4m)³ - (7n)³
= (4m + 7n) {(4m)² + (4m)(7n) + (7n)²}
= (4m + 7n) (16m² + 28mn + 49n)² 

Factorise : 27x³ + y³ + z³ - 9xyz

Answer

27x³ + y³ + z³ - 9xyz = (3x)³ + y³ + z³ - 3×3xyz
Using identity, x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - xz)
27x³ + y³ + z³ - 9xyz
= (3x + y + z) {(3x)² + y² + z² - 3xy - yz - 3xz}
= (3x + y + z) (9x² + y² + z² - 3xy - yz - 3xz)

Verify that: x³ + y³ + z³ - 3xyz = 1/2(x + y + z) [(x - y)² + (y - z)² + (z - x)²]

Answer

We know that,
x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - xz)
⇒ x³ + y³ + z³ - 3xyz = 1/2×(x + y + z) 2(x² + y² + z² - xy - yz - xz)
= 1/2(x + y + z) (2x² + 2y² + 2z² - 2xy - 2yz - 2xz) 
= 1/2(x + y + z) [(x² + y² -2xy) + (y² + z² - 2yz) + (x² + z² - 2xz)] 
= 1/2(x + y + z) [(x - y)² + (y - z)² + (z - x)²]

If x + y + z = 0, show that x³ + y³ + z³ = 3xyz.

Answer

We know that,
x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - xz)
Now put (x + y + z) = 0,  
x³ + y³ + z³ - 3xyz = (0)(x² + y² + z² - xy - yz - xz) 
⇒ x³ + y³ + z³ - 3xyz = 0

Without actually calculating the cubes, find the value of each of the following:
     (i) (-12)³ + (7)³ + (5)³
     (ii) (28)³ + (–15)³ + (-13)³

Answer

(i) (-12)³ + (7)³ + (5)³
 Let x = -12, y = 7 and z = 5
We observed that, x + y + z = -12 + 7 + 5 = 0

We know that if,
x + y + z = 0, then x³ + y³ + z³ = 3xyz
(-12)³ + (7)³ + (5)³ = 3(-12)(7)(5) = -1260

(ii) (28)³ + (–15)³ + (-13)³
 Let x = 28, y = -15 and z = -13
We observed that, x + y + z = 28 - 15 - 13 = 0

We know that if,
x + y + z = 0, then x³ + y³ + z³ = 3xyz
(28)³ + (–15)³ + (-13)³ = 3(28)(-15)(-13) = 16380

Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given: 
(i) Area : 25a² - 35a + 12
(ii) Area : 35 y² + 13y - 12

Answer


(i) Area : 25a² - 35a + 12

Since, area is product of length and breadth therefore by factorizing the given area, we can know the length and breadth of rectangle.
25a² - 35a + 12
= 25a² - 15a -20a + 12
= 5a(5a - 3) - 4(5a - 3)
= (5a - 4)(5a - 3)
Possible expression for length = 5a - 4
Possible expression for breadth = 5a - 3

(ii) Area : 35 y² + 13y - 12
35 y² + 13y - 12
= 35y² - 15y + 28y - 12
= 5y(7y - 3) + 4(7y - 3)
= (5y + 4)(7y - 3)
Possible expression for length = (5y + 4)
Possible expression for breadth = (7y - 3)