If ` x + y = z` then the expression ` x^(3) + y^(3) - z^(3) + 3xyz ` will be equal to

Options

A. 0
B. 3xyz
C. `-3xyz`
D. `z^(3)`

Answer

To solve the expression \( x^3 + y^3 - z^3 + 3xyz \) given that \( x + y = z \), we can follow these steps:

### Step 1: Substitute \( z \) in the expression
Since we know that \( z = x + y \), we can substitute \( z \) into the expression:
\[ x^3 + y^3 - (x + y)^3 + 3xyz \]

### Step 2: Expand \( (x + y)^3 \)
Using the binomial expansion, we have:
\[ (x + y)^3 = x^3 + y^3 + 3x^2y + 3xy^2 \]
Thus, we can rewrite the expression as:
\[ x^3 + y^3 - (x^3 + y^3 + 3x^2y + 3xy^2) + 3xyz \]

### Step 3: Simplify the expression
Now, we can simplify the expression:
\[ x^3 + y^3 - x^3 - y^3 - 3x^2y - 3xy^2 + 3xyz \]
The \( x^3 \) and \( y^3 \) terms cancel out:
\[ -3x^2y - 3xy^2 + 3xyz \]

### Step 4: Factor out common terms
Notice that we can factor out \( 3xy \):
\[ 3xy(-x - y + z) \]
Since \( z = x + y \), we can substitute back:
\[ 3xy(-x - y + (x + y)) = 3xy(0) \]

### Step 5: Conclude the result
Thus, the entire expression simplifies to:
\[ 0 \]

### Final Answer
The value of the expression \( x^3 + y^3 - z^3 + 3xyz \) is \( 0 \).

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