`(1)/(2) (a + b) (a ^(2) + b ^(2) ) +(1)/(2) (a - b) (a ^(2) - b ^(2))` is equal to

Options

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

Answer

To solve the expression \(\frac{1}{2} (a + b) (a^2 + b^2) + \frac{1}{2} (a - b) (a^2 - b^2)\), we will follow these steps:

### Step 1: Expand the first term
We start by expanding \(\frac{1}{2} (a + b) (a^2 + b^2)\).

\[ \frac{1}{2} (a + b) (a^2 + b^2) = \frac{1}{2} \left( a(a^2 + b^2) + b(a^2 + b^2) \right) \]

Expanding this gives:

\[ = \frac{1}{2} (a^3 + ab^2 + b a^2 + b^3) = \frac{1}{2} (a^3 + b^3 + ab^2 + ba^2) \]

### Step 2: Expand the second term
Next, we expand \(\frac{1}{2} (a - b) (a^2 - b^2)\).

Using the difference of squares, we have:

\[ \frac{1}{2} (a - b) (a^2 - b^2) = \frac{1}{2} (a - b)(a - b)(a + b) = \frac{1}{2} (a^3 - b^3) \]

### Step 3: Combine the two expanded terms
Now we combine both expanded terms:

\[ \frac{1}{2} (a^3 + b^3 + ab^2 + ba^2) + \frac{1}{2} (a^3 - b^3) \]

This simplifies to:

\[ = \frac{1}{2} (2a^3 + ab^2 + ba^2) = \frac{1}{2} (2a^3 + 2b^3) \]

### Step 4: Simplify the expression
Now, we can simplify further:

\[ = \frac{1}{2} \cdot 2(a^3 + b^3) = a^3 + b^3 \]

### Final Result
Thus, the expression simplifies to:

\[ \boxed{a^3 + b^3} \]