Q. 115.0( 1 Vote )

# Simplify:

(i) (2x + y + 4z)(4x^{2} + y^{2} + 16z^{2} – 2xy – 4yz – 8zx)

(ii) (x – 3y – 5z)(x^{2} + 9y^{2} + 25z^{2} + 3xy – 15yz + 5zx)

Answer :

(i) (2x + y + 4z) (4x^{2} + y^{2} + 16z^{2} – 2xy – 4yz – 8zx)

⇒ Here the identity used is

a^{3} + b^{3} + c^{3} – 3abc = (a + b + c) (a^{2} + b^{2} + c^{2} – ab – bc – ac)

Comparing this with given condition

(2x + y + 4z) (4x^{2} + y^{2} + 16z^{2} – 2xy – 4yz – 8zx) = (2x)^{3} + y^{3} + (4z)^{3} – 3 × 2x × y × 4z

⇒ 8x^{3} + y^{3} + 64z^{3} – 24xyz

(ii) (x – 3y – 5z)(x^{2} + 9y^{2} + 25z^{2} + 3xy – 15yx + 5zx)

⇒ Here the identity used is

a^{3} + b^{3} + c^{3} – 3abc = (a + b + c) (a^{2} + b^{2} + c^{2} – ab – bc – ac)

Comparing this with given condition

(x – 3y – 5z)(x^{2} + 9y^{2} + 25z^{2} + 3xy – 15yx + 5zx)

= x^{3} + (– 3y)^{3} + (– 5z)^{3} – 3(x) (– 3y)(– 5z)

= x^{3} – 27y^{3} – 125z^{3} – 45xyz

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