In the expansion of `(2x+y)^(3)-(2x-y)^(3)`, the coefficient of `x^(2)y` is :

Options

A. 24
B. 18
C. 12
D. 16

Answer

To find the coefficient of \( x^2y \) in the expression \( (2x+y)^3 - (2x-y)^3 \), we can follow these steps:

### Step 1: Expand the expression using the formula for the difference of cubes.
The formula for the difference of cubes is:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Here, let \( a = (2x + y) \) and \( b = (2x - y) \).

### Step 2: Calculate \( a - b \).
\[ a - b = (2x + y) - (2x - y) = 2y \]

### Step 3: Calculate \( a^2 + ab + b^2 \).
First, we need to find \( a^2 \), \( ab \), and \( b^2 \):
- \( a^2 = (2x + y)^2 = 4x^2 + 4xy + y^2 \)
- \( b^2 = (2x - y)^2 = 4x^2 - 4xy + y^2 \)
- \( ab = (2x + y)(2x - y) = 4x^2 - y^2 \)

Now, combine these:
\[ a^2 + ab + b^2 = (4x^2 + 4xy + y^2) + (4x^2 - y^2) + (4x^2 - 4xy + y^2) \]
\[ = 4x^2 + 4xy + y^2 + 4x^2 - y^2 + 4x^2 - 4xy + y^2 \]
\[ = 12x^2 + 0xy + y^2 = 12x^2 + y^2 \]

### Step 4: Combine everything.
Now substitute back into the difference of cubes formula:
\[ (2x+y)^3 - (2x-y)^3 = (2y)(12x^2 + y^2) \]
\[ = 24yx^2 + 2y^3 \]

### Step 5: Identify the coefficient of \( x^2y \).
From the expression \( 24yx^2 + 2y^3 \), we can see that the coefficient of \( x^2y \) is \( 24 \).

### Final Answer:
The coefficient of \( x^2y \) in the expression \( (2x+y)^3 - (2x-y)^3 \) is **24**.

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