When a + b + c + 3a1/3 b2/3 + 3a2/3 b1/3 is divided by a1/3 + b1/3 + c

1.	When a + b + c + 3a1/3 b2/3 + 3a2/3 b1/3 is divided by a1/3 + b1/3 + c1/3, what is the remainder ?(a) 3a (b) 3b (c) 0 (d) c2/3
Answer» (c) 0 Let a^1/3 = x, b^1/3 = y, c^1/3 = z. Then, a + b + c + 3a^1/3 b^2/3 + 3a^2/3 b^1/3 = x³ + y³ + z³ + 3xy² + 3x²y and a^1/3 + b^1/3 + c^1/3 = x + y + z. Now x³ + y³ + z³ + 3xy² + 3x²y = x³ + 3xy² + 3x²y y³ + z³ = (x + y)³ + z³ = [x + y + z] [(x + y)² - (x + y)z + z²]. ∴ Given expression is completely divisible by (x + y + z), i.e., by a^1/3 + b^1/3+ c^1/3.

When a + b + c + 3a1/3 b2/3 + 3a2/3 b1/3 is divided by a1/3 + b1/3 + c1/3, what is the remainder ?(a) 3a (b) 3b (c) 0 (d) c2/3

Answer»

(c) 0

Let a^1/3 = x, b^1/3 = y, c^1/3 = z. Then,

a + b + c + 3a^1/3 b^2/3 + 3a^2/3 b^1/3 = x³ + y³ + z³ + 3xy² + 3x²y

and a^1/3 + b^1/3 + c^1/3 = x + y + z.

Now x³ + y³ + z³ + 3xy² + 3x²y = x³ + 3xy² + 3x²y y³ + z³

= (x + y)³ + z³ = [x + y + z] [(x + y)² - (x + y)z + z²].

∴ Given expression is completely divisible by (x + y + z),

i.e., by a^1/3 + b^1/3+ c^1/3.