which of the following is a factor of `(x+y)^(3)-(x^(3)+y^(3))`?

Options

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

Explanation

To solve the problem of finding a factor of \((x+y)^{3} - (x^{3} + y^{3})\), we can follow these steps:

### Step 1: Use the identity for the sum of cubes
We know that \(x^{3} + y^{3}\) can be factored using the identity:
\[ x^{3} + y^{3} = (x+y)(x^{2} - xy + y^{2}) \]
Thus, we can rewrite the expression:
\[ (x+y)^{3} - (x^{3} + y^{3}) = (x+y)^{3} - (x+y)(x^{2} - xy + y^{2}) \]

### Step 2: Factor out \((x+y)\)
Now, we can factor out \((x+y)\) from the expression:
\[ = (x+y) \left( (x+y)^{2} - (x^{2} - xy + y^{2}) \right) \]

### Step 3: Expand \((x+y)^{2}\)
Next, we expand \((x+y)^{2}\):
\[ (x+y)^{2} = x^{2} + 2xy + y^{2} \]
So, we substitute this back into the expression:
\[ = (x+y) \left( (x^{2} + 2xy + y^{2}) - (x^{2} - xy + y^{2}) \right) \]

### Step 4: Simplify the expression inside the parentheses
Now, simplify the expression inside the parentheses:
\[ = (x+y) \left( x^{2} + 2xy + y^{2} - x^{2} + xy - y^{2} \right) \]
Notice that \(x^{2}\) and \(-x^{2}\) cancel out, and \(y^{2}\) and \(-y^{2}\) also cancel out:
\[ = (x+y) \left( 2xy + xy \right) \]
This simplifies to:
\[ = (x+y)(3xy) \]

### Conclusion
Thus, the expression \((x+y)^{3} - (x^{3} + y^{3})\) can be factored as:
\[ (x+y)(3xy) \]
Therefore, one of the factors of the expression is \((x+y)\) and another factor is \(3xy\).