`a^3(b-c)^3+b^3(c-a)^3+c^3(a-b)^3`

Options

A. `3abc`
B. `3abc(b-c)(c-a)(a-b)`
C. `3(b-c)(c-a)(a-b)`
D. none of these

Explanation

To simplify the expression \( a^3(b-c)^3 + b^3(c-a)^3 + c^3(a-b)^3 \), we can follow these steps:

### Step 1: Recognize the structure of the expression
The expression consists of three terms, each involving a cube of a variable multiplied by the cube of a difference of the other two variables.

### Step 2: Factor out common terms
Notice that each term can be factored in a way that highlights the symmetry. We can rewrite the expression as:
\[ a^3(b-c)^3 + b^3(c-a)^3 + c^3(a-b)^3 \]

### Step 3: Use the identity for sum of cubes
We can use the identity for the sum of cubes:
\[ x^3 + y^3 + z^3 - 3xyz = (x+y+z)(x^2 + y^2 + z^2 - xy - xz - yz) \]
where \( x = a(b-c) \), \( y = b(c-a) \), and \( z = c(a-b) \).

### Step 4: Set up the expression for \( x+y+z \)
Calculate \( x+y+z \):
\[ x + y + z = a(b-c) + b(c-a) + c(a-b) \]
This simplifies to:
\[ = ab - ac + bc - ab + ca - bc = 0 \]

### Step 5: Apply the identity
Since \( x + y + z = 0 \), we can apply the identity:
\[ x^3 + y^3 + z^3 = 3xyz \]
Thus, we have:
\[ a^3(b-c)^3 + b^3(c-a)^3 + c^3(a-b)^3 = 3(a(b-c))(b(c-a))(c(a-b)) \]

### Step 6: Final expression
Now we can write the final expression as:
\[ = 3abc(b-c)(c-a)(a-b) \]

### Final Answer
The simplified expression is:
\[ 3abc(b-c)(c-a)(a-b) \]

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