Question

Find the following product:
(i) (3x+2y+2z) (9x²+4y²+4z²-6xy-4yz-6zx)
(ii) (4x-3y+2z) (16x²+9y²+4z²+12xy+6yz-8zx)
(iii) (2a-3b-2c) (4a²+9b²+4c²+6ab-6bc+4ca)
(iv) (3x-4y+5z)(9x²+16y²+25z²+12xy-15zx+20yz)

Answer 1

Concept :  

We know that (a + b + c) (a² + b² + c² - ab - bc - ca) = a³ + b³ + c³ - 3abc

We apply this identity in all the questions here :  

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

= (3x + 2y + 2z) [(3x)² + (2y) ² + (2z)² – 3x x 2y + 2y x 2z + 2z x 3x]

= (3x)³ + (2y)³+ (2z)³ – 3 x 3x x 2y x 2z

= 27x³ + 8y³ + 8Z³ – 36xyz

 

(ii) (4x - 3y + 2z) (16x² + 9y² + 4z² + 12xy + 6yz - 8zx)

= (4x -3y + 2z) [(4x)² + (-3y) ² + (2z) ²– 4x x (-3y + (3y) x (2z) – (2z x 4x)]

= (4x)³  + (-3y)³  + (2z) ³ – 3 x 4x x (-3y) x (2z)

= 64x³ – 27y³ + 8z³ + 72xyz

 

(iii) (2a - 3b - 2c) (4a² + 9b² + 4c² + 6ab - 6bc + 4ca)

= (2a -3b- 2c) [(2a)² + (3b)² + (2c)² – 2a x (-3b) – (-3b) x (-2c) – (-2c) x 2a]

= (2a)³ + (3b) ³ + (-2c) ³ -3x 2a x (-3 b) (-2c)

= 8a³ – 21b³ – 8c³ – 36abc

 

(iv) (3x - 4y + 5z)(9x² + 16y² + 25z² + 12xy - 15zx + 20yz)

= [3x + (-4y) + 5z] [(3x)² + (-4y)² + (5z) ² – 3x x (-4y) -(-4y) (5z) – 5z x 3x]

= (3x) ³ + (-4y) ³ + (5z)³  – 3 x 3x x (-4y) (5z)

= 27x³  – 64y³  + 125z³ + 180xyz

HOPE THIS ANSWER WILL HELP YOU…..

 

Some questions of this chapter :

If x+y+z = 8 and xy+yz+zx =20, find the value of x³ +y³ +z³ -3xyz.

https://brainly.in/question/15897798

 

If a+b+c=9 and ab+bc+ca=26, find the value of a³ +b³ +c³ -3abc.

https://brainly.in/question/15897805

Answer 2

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(i) (3x + 2y + 2z) (9x2 + 4y2 + 4z2 – 6xy – 4yz – 6zx) = (3x + 2y + 2z) [(3x)2 + (2y)2 + (2z)2 – 3x x 2y + 2y x 2z + 2z x 3x] = (3x)3 + (2y)3 + (2z)3 – 3 x 3x x 2y x 2z = 27x3 + 8y3 + 8Z3 – 36xyz

(ii) (4x – 3y + 2z) (16x2 + 9y2 + 4z2 + 12xy + 6yz – 8zx) = (4x -3y + 2z) [(4x)2 + (-3y)2 + (2z)2 – 4x x (-3y + (3y) x (2z) – (2z x 4x)] = (4x)3 + (-3y)3 + (2z)3 – 3 x 4x x (-3y) x (2z) = 64x3 – 27y3 + 8z3 + 72xyz

(iii) (2a -3b- 2c) (4a2 + 9b2 + 4c2 + 6ab – 6bc + 4ca) = (2a - 3b - 2c) [(2a)2 + (3b)2 + (2c)2 – 2a x (-3b) – (-3b) x (-2c) – (-2c) x 2a] = (2a)3 + (3b)3 + (-2c)3 -3 x 2a x (-3b) (-2c) = 8a3 – 21b3 – 8c3 – 36abc

(iv) (3x – 4y + 5z) (9x2 + 16y2 + 25z2 + 12xy – 15zx + 20yz) = [3x + (-4y) + 5z] [(3x)2 + (-4y)2 + (5z)2 – 3x x (-4y) -(-4y) (5z) – 5z x 3x] = (3x)3 + (-4y)3 + (5z)3 – 3 x 3x x (-4y) (5z) = 27x3 – 64y3 + 125z3 +