If ` x + y = 4 ` then what is the value of ` x^(3) + y^(3) + 12 xy`

Options

A. 16
B. 32
C. 64
D. 256

Answer

To solve the problem, we need to find the value of \( x^3 + y^3 + 12xy \) given that \( x + y = 4 \).

### Step-by-step Solution:

1. **Start with the given equation**:
\[ x + y = 4 \]

2. **Use the identity for the sum of cubes**:
The formula for the sum of cubes is:
\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]
We can also express \( x^2 + y^2 \) in terms of \( x + y \) and \( xy \):
\[ x^2 + y^2 = (x + y)^2 - 2xy \]

3. **Substituting \( x + y \) into the identity**:
First, we need to find \( x^2 + y^2 \):
\[ x^2 + y^2 = (x + y)^2 - 2xy = 4^2 - 2xy = 16 - 2xy \]

4. **Substituting into the sum of cubes formula**:
Now, substituting \( x^2 + y^2 \) into the sum of cubes:
\[ x^3 + y^3 = (x + y)((x^2 + y^2) - xy) = 4((16 - 2xy) - xy) = 4(16 - 3xy) \]
Simplifying this gives:
\[ x^3 + y^3 = 64 - 12xy \]

5. **Now, substitute \( x^3 + y^3 \) into the expression we need to evaluate**:
We need to find \( x^3 + y^3 + 12xy \):
\[ x^3 + y^3 + 12xy = (64 - 12xy) + 12xy \]
The \( -12xy \) and \( +12xy \) cancel out:
\[ x^3 + y^3 + 12xy = 64 \]

### Final Answer:
Thus, the value of \( x^3 + y^3 + 12xy \) is:
\[ \boxed{64} \]