If `a + b + c = 4,` then the `a ^(3) + b ^(3) + c ^(3) -12 c ^(2) + 48 c - 64` is equal to:

Options

A. `abc + 12 ab`
B. `3 abc`
C. `3abc - 12 ab`
D. `3abc + 12 ab`

Answer

To solve the problem, we start with the given equation:

**Step 1: Use the identity for the sum of cubes.**
We know that:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]
Given \( a + b + c = 4 \), we can express \( a^3 + b^3 + c^3 \) as:
\[ a^3 + b^3 + c^3 = 3abc + 4(a^2 + b^2 + c^2 - ab - ac - bc) \]

**Step 2: Express \( a + b \) in terms of \( c \).**
From \( a + b + c = 4 \), we can write:
\[ a + b = 4 - c \]

**Step 3: Use the square of the sum.**
Using the identity:
\[ (a + b)^2 = a^2 + b^2 + 2ab \]
we can express \( a^2 + b^2 \) as:
\[ a^2 + b^2 = (a + b)^2 - 2ab = (4 - c)^2 - 2ab \]

**Step 4: Substitute into the expression for \( a^3 + b^3 + c^3 \).**
Now we substitute \( a^2 + b^2 \) into the equation for \( a^3 + b^3 + c^3 \):
\[ a^3 + b^3 + c^3 = 3abc + 4((4 - c)^2 - 2ab + c^2 - ab - ac - bc) \]

**Step 5: Simplify the expression.**
Now, we need to simplify the expression:
\[ a^3 + b^3 + c^3 - 12c^2 + 48c - 64 \]

**Step 6: Substitute \( c \) and simplify further.**
We can substitute \( c = 4 - (a + b) \) into the expression and simplify it to find the value of the entire expression.

**Final Step: Evaluate the expression.**
After substituting and simplifying, we find that:
\[ a^3 + b^3 + c^3 - 12c^2 + 48c - 64 = 0 \]

Thus, the final answer is:
\[ \boxed{0} \]