What must be sutracted from `a ^(3) - 3a ^(2) b + 3 ab ^(2) - b ^(3)` get `a ^(3) - b ^(3)` ?

Options

A. `- 3a ^(2) b + 3 ab ^(2)`
B. `3a^(2) b - 3ab ^(2)`
C. `a ^(3) - b ^(3)`
D. ` - a^(3) + b ^(3)`

Answer

To solve the problem, we need to determine what must be subtracted from the expression \( a^3 - 3a^2b + 3ab^2 - b^3 \) in order to obtain \( a^3 - b^3 \).

### Step-by-Step Solution:

1. **Set Up the Equation**:
We start with the expression:
\[ a^3 - 3a^2b + 3ab^2 - b^3 \]
We want to subtract some value \( x \) from this expression to equal \( a^3 - b^3 \):
\[ a^3 - 3a^2b + 3ab^2 - b^3 - x = a^3 - b^3 \]

2. **Rearranging the Equation**:
Rearranging the equation gives us:
\[ - x = (a^3 - b^3) - (a^3 - 3a^2b + 3ab^2 - b^3) \]

3. **Simplifying the Right Side**:
Now, simplify the right side:
\[ - x = a^3 - b^3 - a^3 + 3a^2b - 3ab^2 + b^3 \]
This simplifies to:
\[ - x = 3a^2b - 3ab^2 \]

4. **Solving for \( x \)**:
To find \( x \), we multiply both sides by -1:
\[ x = - (3a^2b - 3ab^2) \]
Which simplifies to:
\[ x = -3a^2b + 3ab^2 \]

5. **Final Expression**:
Therefore, the value that must be subtracted is:
\[ x = -3a^2b + 3ab^2 \]

### Conclusion:
The answer is \( -3a^2b + 3ab^2 \).