If ` x + y = a and xy = b^(2)` then the value of ` x^(3) - x^(2) y - xy^(2) + y^(3)` in terms of a and b is

Options

A. `( a^(2) + 4 b ^(2)) a `
B. ` a^(3) - 3 b ^(2)`
C. ` a^(3) - 4 b ^(2) a `
D. ` a^(3) + 3 b ^(2)`

Answer

To solve the problem, we need to find the value of \( x^3 - x^2y - xy^2 + y^3 \) in terms of \( a \) and \( b \), given that \( x + y = a \) and \( xy = b^2 \).

### Step-by-Step Solution:

1. **Rewrite the Expression:**
We start with the expression:
\[ x^3 - x^2y - xy^2 + y^3 \]
We can rearrange this expression as follows:
\[ x^3 + y^3 - x^2y - xy^2 \]

2. **Use the Sum of Cubes Formula:**
The sum of cubes can be expressed using the formula:
\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]
Since \( x + y = a \), we substitute:
\[ x^3 + y^3 = a(x^2 - xy + y^2) \]

3. **Express \( x^2 + y^2 \):**
We know that:
\[ x^2 + y^2 = (x + y)^2 - 2xy = a^2 - 2b^2 \]
Therefore, we can substitute this into our expression for \( x^3 + y^3 \):
\[ x^3 + y^3 = a((a^2 - 2b^2) - b^2) = a(a^2 - 3b^2) \]

4. **Combine the Terms:**
Now we need to simplify \( -x^2y - xy^2 \):
\[ -x^2y - xy^2 = -xy(x + y) = -b^2a \]

5. **Final Expression:**
Combining everything, we have:
\[ x^3 - x^2y - xy^2 + y^3 = a(a^2 - 3b^2) - b^2a \]
Simplifying this gives:
\[ = a^3 - 3ab^2 - ab^2 = a^3 - 4ab^2 \]

Thus, the final result is:
\[ \boxed{a^3 - 4ab^2} \]