If `8(x+y)^(3)-27(x-y)^(3)=(5y-x)(Ax^(2)+By^(2)+Cxy)` then find `A+B-C`.

Options

A. 32
B. 18
C. 26
D. 36

Answer

To solve the equation \( 8(x+y)^3 - 27(x-y)^3 = (5y-x)(Ax^2 + By^2 + Cxy) \) and find \( A + B - C \), we will follow these steps:

### Step 1: Expand the Left-Hand Side (LHS)

We start with the left-hand side of the equation:
\[ 8(x+y)^3 - 27(x-y)^3 \]

Using the binomial expansion:
\[ a^3 - b^3 = (a-b)(a^2 + ab + b^2) \]
where \( a = 2(x+y) \) and \( b = 3(x-y) \).

Thus, we can rewrite:
\[ 8(x+y)^3 = (2(x+y))^3 = 2^3(x+y)^3 = 8(x+y)^3 \]
\[ 27(x-y)^3 = (3(x-y))^3 = 27(x-y)^3 \]

Now, we can express the LHS as:
\[ = (2(x+y) - 3(x-y))((2(x+y))^2 + 2(x+y)(3(x-y)) + (3(x-y))^2) \]

### Step 2: Simplify \( 2(x+y) - 3(x-y) \)

Calculating \( 2(x+y) - 3(x-y) \):
\[ = 2x + 2y - 3x + 3y = -x + 5y \]

### Step 3: Calculate \( (2(x+y))^2 + 2(x+y)(3(x-y)) + (3(x-y))^2 \)

Calculating each term:
1. \( (2(x+y))^2 = 4(x+y)^2 = 4(x^2 + 2xy + y^2) = 4x^2 + 8xy + 4y^2 \)
2. \( 2(x+y)(3(x-y)) = 6(x+y)(x-y) = 6(x^2 - y^2) = 6x^2 - 6y^2 \)
3. \( (3(x-y))^2 = 9(x-y)^2 = 9(x^2 - 2xy + y^2) = 9x^2 - 18xy + 9y^2 \)

Now, combine these:
\[ 4x^2 + 8xy + 4y^2 + 6x^2 - 6y^2 + 9x^2 - 18xy + 9y^2 \]
Combine like terms:
\[ (4x^2 + 6x^2 + 9x^2) + (4y^2 - 6y^2 + 9y^2) + (8xy - 18xy) \]
\[ = 19x^2 + 7y^2 - 10xy \]

### Step 4: Combine LHS

Now, substituting back into the LHS:
\[ LHS = (-x + 5y)(19x^2 + 7y^2 - 10xy) \]

### Step 5: Compare with Right-Hand Side (RHS)

The RHS is given as:
\[ (5y - x)(Ax^2 + By^2 + Cxy) \]

From our LHS, we can identify:
- \( A = 19 \)
- \( B = 7 \)
- \( C = -10 \)

### Step 6: Calculate \( A + B - C \)

Now, we calculate:
\[ A + B - C = 19 + 7 - (-10) = 19 + 7 + 10 = 36 \]

### Final Answer

Thus, the value of \( A + B - C \) is:
\[ \boxed{36} \]