What should be added to `x^3 + 3x^2y + 3xy^2 + y^3 " to get " x^3 + y^3` ?

Options

A. `-3x^2y - 3xy^2`
B. `-3x^2y + 3xy^2`
C. `3x^2y - 3xy^2`
D. `-2x^3 - 2y^3 + 3x^2y + 3xy^2`

Answer

To find out what should be added to the expression \( x^3 + 3x^2y + 3xy^2 + y^3 \) to get \( x^3 + y^3 \), we can follow these steps:

### Step 1: Write down the equation
We need to find a value \( k \) such that:
\[ x^3 + 3x^2y + 3xy^2 + y^3 + k = x^3 + y^3 \]

### Step 2: Rearrange the equation
To isolate \( k \), we can rearrange the equation:
\[ k = (x^3 + y^3) - (x^3 + 3x^2y + 3xy^2 + y^3) \]

### Step 3: Simplify the right-hand side
Now, let's simplify the right-hand side:
\[ k = x^3 + y^3 - x^3 - 3x^2y - 3xy^2 - y^3 \]

### Step 4: Cancel out like terms
Notice that \( x^3 \) and \( -x^3 \) cancel each other out, and \( y^3 \) and \( -y^3 \) also cancel each other out:
\[ k = -3x^2y - 3xy^2 \]

### Step 5: Factor out common terms
We can factor out \(-3xy\) from the expression:
\[ k = -3xy(x + y) \]

### Final Answer
Thus, the value that should be added to \( x^3 + 3x^2y + 3xy^2 + y^3 \) to get \( x^3 + y^3 \) is:
\[ -3xy(x + y) \]