Express each of the following in factors form,
` a^(3)(b- c) ^(3) +b^(3) (c-a) ^(3)+ c^(3) (a-b) "^(3)`

Answer

To express the given expression \( a^3(b - c)^3 + b^3(c - a)^3 + c^3(a - b)^3 \) in factor form, we can follow these steps:

### Step 1: Identify the terms
We start with the expression:
\[ a^3(b - c)^3 + b^3(c - a)^3 + c^3(a - b)^3 \]

### Step 2: Rewrite the expression
We can rewrite the expression using the property of cubes:
\[ = a(b - c)^3 + b(c - a)^3 + c(a - b)^3 \]

### Step 3: Define new variables
Let:
\[ x = a(b - c), \quad y = b(c - a), \quad z = c(a - b) \]

### Step 4: Check if \( x + y + z = 0 \)
Now, we need to check if \( x + y + z = 0 \):
\[ x + y + z = a(b - c) + b(c - a) + c(a - b) \]

### Step 5: Expand and simplify
Expanding the expression:
\[ = ab - ac + bc - ab + ac - bc \]
Here, we can see that all terms cancel out:
\[ = 0 \]

### Step 6: Apply the identity
Since \( x + y + z = 0 \), we can use the identity:
\[ x^3 + y^3 + z^3 = 3xyz \]
Thus, we have:
\[ a^3(b - c)^3 + b^3(c - a)^3 + c^3(a - b)^3 = 3xyz \]

### Step 7: Substitute back the values of \( x, y, z \)
Substituting back:
\[ = 3 \cdot a(b - c) \cdot b(c - a) \cdot c(a - b) \]

### Step 8: Factor out common terms
We can factor this expression:
\[ = 3abc(b - c)(c - a)(a - b) \]

### Final Answer
Thus, the expression in factor form is:
\[ 3abc(b - c)(c - a)(a - b) \]

---