If (x - y) = 3, then what is the value of ` ( x^(3) - y^(2) - 9 xy)` ?

Options

A. (A) 3
B. (B)9
C. (C)18
D. (D)27

Answer

To solve the problem, we need to find the value of \( x^3 - y^2 - 9xy \) given that \( x - y = 3 \).

### Step 1: Express \( x \) in terms of \( y \)
From the equation \( x - y = 3 \), we can express \( x \) as:
\[ x = y + 3 \]

**Hint:** Rearranging equations can help express one variable in terms of another.

### Step 2: Substitute \( x \) into the expression
Now, we substitute \( x = y + 3 \) into the expression \( x^3 - y^2 - 9xy \):
\[ x^3 = (y + 3)^3 \]
Expanding \( (y + 3)^3 \) using the binomial theorem:
\[ (y + 3)^3 = y^3 + 3 \cdot (y^2 \cdot 3) + 3 \cdot (y \cdot 3^2) + 3^3 = y^3 + 9y^2 + 27y + 27 \]
So, we have:
\[ x^3 = y^3 + 27y + 27 \]

**Hint:** Remember to use the binomial expansion to simplify expressions.

### Step 3: Substitute \( x \) into the expression
Now we also need to substitute \( x \) into the term \( -9xy \):
\[ -9xy = -9(y + 3)y = -9y^2 - 27y \]

**Hint:** When substituting, distribute the multiplication carefully.

### Step 4: Combine all parts of the expression
Now we can combine everything in the expression:
\[ x^3 - y^2 - 9xy = (y^3 + 27y + 27) - y^2 - (9y^2 + 27y) \]
This simplifies to:
\[ y^3 + 27y + 27 - y^2 - 9y^2 - 27y = y^3 - 10y^2 + 27 \]

**Hint:** Combine like terms carefully to simplify the expression.

### Step 5: Evaluate the expression
Now we can evaluate this expression. Since we don't have a specific value for \( y \), we can choose any value that satisfies \( x - y = 3 \). Let's choose \( y = 1 \):
\[ x = 1 + 3 = 4 \]
Substituting \( y = 1 \) into the expression:
\[ 1^3 - 10(1^2) + 27 = 1 - 10 + 27 = 18 \]

**Hint:** Always check if the values satisfy the original equation before substituting.

### Final Result
Thus, the value of \( x^3 - y^2 - 9xy \) is:
\[ \boxed{18} \]