Factorise `a^(3)-(b-c)^(3)`.

Answer

To factorize the expression \( a^3 - (b - c)^3 \), we can use the formula for the difference of cubes, which states:

\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \]

In our case, we can identify:
- \( x = a \)
- \( y = b - c \)

Now, let's apply the formula step by step.

### Step 1: Identify \( x \) and \( y \)
We have:
- \( x = a \)
- \( y = b - c \)

### Step 2: Apply the difference of cubes formula
Using the formula, we can write:

\[ a^3 - (b - c)^3 = (a - (b - c))(a^2 + a(b - c) + (b - c)^2) \]

### Step 3: Simplify the first factor
The first factor simplifies as follows:

\[ a - (b - c) = a - b + c \]

### Step 4: Expand the second factor
Now, we need to expand the second factor:

\[ a^2 + a(b - c) + (b - c)^2 \]

Calculating each term:
- The first term is \( a^2 \).
- The second term is \( ab - ac \).
- The third term, using the expansion \( (b - c)^2 = b^2 - 2bc + c^2 \).

So, we have:

\[ a^2 + ab - ac + b^2 - 2bc + c^2 \]

### Step 5: Combine all parts
Now, we can combine everything together:

\[ a^3 - (b - c)^3 = (a - b + c)(a^2 + ab - ac + b^2 - 2bc + c^2) \]

### Final Answer
Thus, the factorized form of \( a^3 - (b - c)^3 \) is:

\[ (a - b + c)(a^2 + ab - ac + b^2 - 2bc + c^2) \]

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