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USE CODE: SELF10

Chapter 5: Factorization of Algebraic Expressions Exercise – 5.4

Question: 1

Find the value

a3 + 8b3 + 64c3 − 24abc

Solution:

= (a)3 + (2b)3 + (4c)3 − 3 × a × 2b × 4c

= (a + 2b + 4c)(a2 + (2b)2 + (4c)2 − a × 2b − 2b × 4c − 4c × a)

[∴ a3 + b3 + c3 − 3abc = (a + b + c)(a2 + b2 + c2 − ab − bc − ca)]

= (a + 2b + 4c)(a2 + 4b2 + 16c2 − 2ab − 8bc − 4ac)

∴ a3 + 8b3 + 64c3 − 24abc 

= (a + 2b + 4c)(a2 + 4b2 + 16c2 − 2ab − 8bc − 4ac)

 

Question: 2

Find the value

x3 − 8y3 + 27z3 + 18xyz

Solution:

= x3 − (2y)3 + (3z)3 − 3 × x × (−2y)(3z)

= (x + (−2y) + 3z)(x2 + (−2y)2 + (3z)2 − x(−2y) − (−2y)(3z) − 3z(x))

[∴ a3 + b3 + c3 − 3abc = (a + b + c)(a2 + b2 + c2 − ab − bc − ca)]

= (x + (−2y) + 3z)(x2 + 4y2 + 9z2 + 2xy + 6yz − 3zx)

∴ x3 − 8y3 + 27z3 + 18xyz

 = (x + (−2y) + 3z)(x2 + 4y2 + 9z2 + 2xy + 6yz − 3zx)

 

Question: 3

Find the value

Solution:

= (x/3)3 + (−y)3 + (5z)3 − 3 × x/3(−y)(5z)

= (x/3 + (−y) + 5z)((x/3)2 + (−y)2 + (5z)2 − x/3(−y) − (−y)5z − 5z(x/3))

= (x/3 − y + 5z)(x2/9 + y2 + 25z2 + xy/3 + 5yz − 5zx/3)

= (x/3 − y + 5z)(x2/9 + y2 + 25z2 + xy/3 + 5yz − 5zx/3)

 

Question: 4

Find the value

8x3 + 27y3 − 216z3 + 108xyz

Solution:

= (2x)3 + (3y)3 + (− 6y)3 − 3(2x)(3y)(−6z)

= (2x + 3y + (− 6z))((2x)2 + (3y)2 + (−6z)2 − 2x × 3y − 3y(−6z) − (−6z)2x)

= (2x + 3y + (− 6z))(4x2 + 9y2 + 36z2 − 6xy + 18yz + 12zx)

? 8x3 + 27y3 − 216z3 + 108xyz 

= (2x + 3y + (− 6z))(4x2 + 9y2 + 36z2 − 6xy + 18yz + 12zx)

 

Question: 5

Find the value

125 + 8x3 − 27y3 + 90xy

Solution:

= (5)3 + (2x)3 + (−3y)3 − 3 × 5 × 2x × (−3y)

= (5 + 2x + (−3y))(52 + (2x)2 + (−3y)2 − 5(2x) − 2x(−3y) − (−3y)5)

= (5 + 2x − 3y)(25 + 4x2 + 9y2 − 10x + 6xy + 15y)

∴ 125 + 8x3 − 27y3 + 90xy 

= (5 + 2x − 3y)(25 + 4x2 + 9y2 − 10x + 6xy + 15y)

 

Question: 6

Find the value

(3x − 2y)3 + (2y − 4z)3 + (4z − 3x)3

Solution:

Let (3x − 2y) = a, (2y − 4z) = b, (4z − 3x) = c

∴ a + b + c = 3x − 2y + 2y − 4z + 4z − 3x = 0

∵ a + b + c = 0 ∴ a3 + b3 + c3 = 3abc

= 3(3x − 2y)(2y − 4z)(4z − 3x)

∴ (3x − 2y)3 + (2y − 4z)3 + (4z − 3x)3 

= 3(3x − 2y)(2y − 4z)(4z − 3x)

 

Question: 7

Find the value

(2x − 3y)3 + (4z − 2x)3 + (3y − 4z)3

Solution:

Let 2x - 3y = a, 4z - 2x = b, 3y - 4z = c

∴ a + b + c = 2x − 3y + 4z − 2x + 3y − 4z = 0

∵ a + b + c = 0 ∴ a3 + b3 + c3 = 3abc

= 3(2x − 3y)(4z − 2x)(3y − 4z)

∴ (2x − 3y)3 + (4z − 2x)3 + (3y − 4z)3 

= 3(2x − 3y)(4z − 2x)(3y − 4z)

 

Question: 8

Find the value

[x/2 + y + z/3]3 + [x/3 − 2y/3 + z]3 + [−5x/6 − y/3 − 4z/3]3

Solution:

Let [x/2 + y + z/3] = a,[x/3 − 2y/3 + z] = b, [−5x/6 − y/3 − 4z/3] = c

a + b + c = x/2 + y + z/3 + x/3 − 2y/3 + z − 5x/6 − y/3 − 4z/3

a + b + c = (x/2 + x/3 − 5x/6) + (y − 2y/3 − y/3) + (z/3 + z − 4z/3)

a + b + c = 3x/6 + 2x/6 − 5x/6 + 3y/3 − 2y/3 − y/3 + z/3 + 3z/3 − 4z/3

a + b + c = 0

∵ a + b + c = 0  ∴  a3 + b3 + c3 = 3abc

= 3(x/2 + y + z/3)(x/3 − 2y/3 + z)(−5x/6 − y/3 − 4z/3)

∴ [x2 + y + z3]3 + [x3 − 2y3 + z]3 + [−5x/6 − y/3 − 4z/3]3 

= 3(x/2 + y + z/3)(x/3 − 2y/3 + z)(−5x/6 − y/3 − 4z/3)

 

Question: 9

Find the value

(a − 3b)3 + (3b − c)3 + (c − a)3

Solution:

Let a - 3b = x, 3b - c = y, c - a = z

x + y + z = a − 3b + 3b − c + c − a = 0

(∴ x + y + z = 0)?

x3 + y3 + z3 = 3xyz

= 3(a − 3b)(3b − c)(c − a)

∴ (a − 3b)3 + (3b − c)3 + (c − a)3 

= 3(a − 3b)(3b − c)(c − a)

 

Question: 10

Find the value

Solution:

 

Question: 11

Find the value

Solution:

 

Question: 12

Find the value

8x3 − 125y3 + 216 + 180xy

Solution:

= (2x)3 + (−5y)3 + 63 − 3 × (2x)(−5y)(6)

= (2x + (−5y) + 6)((2x)2 + (−5y)2 + 62 − 2x × (−5y) − (−5y)6 − 6(2x))

= (2x − 5y + 6)(4x2 + 25y2 + 36 + 10xy + 30y − 12x)

∴ 8x3 − 125y3 + 216 + 180xy 

= (2x − 5y + 6)(4x2 + 25y2 + 36 + 10xy + 30y − 12x)

 

Question: 13

Find the value

Solution:

 

Question: 14

Find the value of x3 + y3 − 12xy + 64 when x + y = - 4.

Solution:

= x3 + y3 + 64 − 12xy

= x3 + y3 + 43 − 3(x)(y)(4)

= (x + y + 4)(x2 + y2 + 42 − xy − y × 4 − 4 × x)

= (−4 + 4)(x2 + y2 + 16 − xy − 4y − 4x)                            

[∴ x + y = −4] = 0

∴  x3 + y3 − 12xy + 64 = 0

 

Question: 15

Multiply:

(i) x2 + y2 + z2 − xy + xz + yz by x + y − z

(ii) x2 + 4y2 + z2 + 2xy  + xz − 2yzbyx − 2y − z

(iii) x2 + 4y2 + 2xy−3x+6y+9  by  (x−2y+3)

(iv) 9x2 + 25y2 + 15xy + 12x − 20y + 16by 3x − 5y + 4

Solution:

(i) x2 + y2 + z2 − xy + xz + yz by x + y − z

= (x2 + y2 + z2 − xy + xz + yz)(x + y − z)

= x3 + y3 + z3 − 3xyz

(ii) x2 + 4y2 + z2 + 2xy  + xz − 2yzbyx − 2y − z

x2 + (−2y)2 + (-z)2 − (−2y)(−z) − (−z)(x) 

= x3 + (−2y)3 + (-z)3 − 3x(−2y)(−z)

⇒ x2 + 4y2 + z2 + 2xy − 2yz + zx

= x3 − 8y3 − z3 − 6xyz

(iii) x2 + 4y2 + 2xy−3x + 6y + 9  by (x − 2y + 3)

(x)2 + (−2y)2 + (3)2 − (x)(−2y) − (−2y)(3) − 3(x)

= (x)3 + (−2y)3 + 33 − 3(x)(−2y)(3)

⇒ x2 + 4y2 + 9 + 2xy + 6y − 3x

= x3 − 8y3 + 27 + 18xy

(iv) 9x2 + 25y2 + 15xy + 12x − 20y + 16by 3x − 5y + 4

(3x)2 + (5y)2 + 42 − (−3x)(5y) − (5y)(4) − (4)(−3x)

= (−3x)3 + (5y)3 + 43 − 3(−3x)(5y)(4)

⇒ 9x2 + 25y2 + 16 + 15xy − 20y + 12x

= −27x3 + 125y3 + 64 + 180xy

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