If `x +y =a` and `x^(2) +y^(2)=b`, then the value of `(x^(3)+y^(3))`, is

Options

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

Answer

To find the value of \( x^3 + y^3 \) given that \( x + y = a \) and \( x^2 + y^2 = b \), we can follow these steps:

### Step 1: Use the identity for \( x^2 + y^2 \)
We know that:
\[ x^2 + y^2 = (x + y)^2 - 2xy \]
Substituting \( x + y = a \):
\[ b = a^2 - 2xy \]

### Step 2: Solve for \( xy \)
Rearranging the equation from Step 1 gives:
\[ 2xy = a^2 - b \]
Thus, we can express \( xy \) as:
\[ xy = \frac{a^2 - b}{2} \]

### Step 3: Use the identity for \( x^3 + y^3 \)
The formula for \( x^3 + y^3 \) is:
\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]
We can rewrite \( x^2 - xy + y^2 \) using the known values:
\[ x^2 + y^2 = b \quad \text{and} \quad xy = \frac{a^2 - b}{2} \]
Thus:
\[ x^2 - xy + y^2 = b - xy = b - \frac{a^2 - b}{2} \]

### Step 4: Simplify \( x^2 - xy + y^2 \)
Substituting \( xy \) into the equation:
\[ x^2 - xy + y^2 = b - \frac{a^2 - b}{2} = b - \frac{a^2}{2} + \frac{b}{2} = \frac{2b - a^2 + b}{2} = \frac{3b - a^2}{2} \]

### Step 5: Substitute back into the formula for \( x^3 + y^3 \)
Now substituting back into the formula for \( x^3 + y^3 \):
\[ x^3 + y^3 = (x + y) \left( x^2 - xy + y^2 \right) = a \cdot \frac{3b - a^2}{2} \]
This simplifies to:
\[ x^3 + y^3 = \frac{a(3b - a^2)}{2} \]

### Final Result
Thus, the value of \( x^3 + y^3 \) is:
\[ \frac{3ab - a^3}{2} \]