The value of `(x-y)^3+(x+y)^3+3(x-y)^2(x+y)+3(x+y)^2(x-y)` is

Options

A. `27x^3`
B. `8(x^3-y^3)`
C. `8x^3y^3`
D. `8x^3`

Answer

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

### Step 1: Group the terms
We can rewrite the expression by grouping similar terms:
\[ (x-y)^3 + (x+y)^3 + 3(x-y)^2(x+y) + 3(x+y)^2(x-y) \]

### Step 2: Factor out common terms
Notice that we can factor out \((x-y)\) and \((x+y)\) from the expression:
\[ = (x-y)^3 + (x+y)^3 + 3(x-y)(x+y)((x-y) + (x+y)) \]

### Step 3: Simplify the expression
We can simplify the expression inside the parentheses:
\[ (x-y) + (x+y) = 2x \]
Thus, we can rewrite the expression as:
\[ = (x-y)^3 + (x+y)^3 + 3(x-y)(x+y)(2x) \]

### Step 4: Use the identity for cubes
We can use the identity \(a^3 + b^3 + 3ab(a + b) = (a + b)^3\). Here, let \(a = (x-y)\) and \(b = (x+y)\):
\[ = (x-y + x+y)^3 = (2x)^3 \]

### Step 5: Calculate the final value
Now, we calculate \((2x)^3\):
\[ (2x)^3 = 8x^3 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{8x^3} \]

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