Solving Algebraic Expansion Problems

The question asks for the value of \(A + B + C\) from the given equation:

\[ (x + y)^3 + 8 (x - y)^3 = (3x + Ay) (3x^2 + Bxy + Cy^2) \]

This equation involves the sum of two cubes. We can rewrite the left side using the identity \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).

Let \(a = (x + y)\) and \(b = 2(x - y)\), since \(8(x - y)^3 = (2(x - y))^3\).

Now, apply the sum of cubes identity to the left side:

\[ (x + y)^3 + (2(x - y))^3 = [(x + y) + 2(x - y)][(x + y)^2 - (x + y)(2(x - y)) + (2(x - y))^2] \]

Step 1: Simplify the first factor \((a + b)\)

The first factor is \((x + y) + 2(x - y)\). Let's simplify this:

\[ (x + y) + 2(x - y) = x + y + 2x - 2y \]

Combine like terms:

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

So, the first factor is \((3x - y)\).

Step 2: Simplify the second factor \((a^2 - ab + b^2)\)

The second factor is \((x + y)^2 - (x + y)(2(x - y)) + (2(x - y))^2\). Let's simplify each term within this factor:

• \((x + y)^2 = x^2 + 2xy + y^2\)
• \((x + y)(2(x - y)) = 2(x + y)(x - y)\). Using the difference of squares identity \((p + q)(p - q) = p^2 - q^2\), this becomes \(2(x^2 - y^2) = 2x^2 - 2y^2\).
• \((2(x - y))^2 = 4(x - y)^2\). Using the square of a difference identity \((p - q)^2 = p^2 - 2pq + q^2\), this becomes \(4(x^2 - 2xy + y^2) = 4x^2 - 8xy + 4y^2\).

Now, substitute these simplified terms back into the second factor expression:

\[ (x^2 + 2xy + y^2) - (2x^2 - 2y^2) + (4x^2 - 8xy + 4y^2) \]

Remove the parentheses, remembering to distribute the negative sign:

\[ x^2 + 2xy + y^2 - 2x^2 + 2y^2 + 4x^2 - 8xy + 4y^2 \]

Combine like terms (\(x^2\), \(xy\), and \(y^2\)):

• \(x^2\) terms: \(x^2 - 2x^2 + 4x^2 = (1 - 2 + 4)x^2 = 3x^2\)
• \(xy\) terms: \(2xy - 8xy = (2 - 8)xy = -6xy\)
• \(y^2\) terms: \(y^2 + 2y^2 + 4y^2 = (1 + 2 + 4)y^2 = 7y^2\)

So, the second factor simplifies to \(3x^2 - 6xy + 7y^2\).

Step 3: Equate the simplified left side with the right side

The simplified left side of the equation is the product of the two factors we found:

\[ (3x - y)(3x^2 - 6xy + 7y^2) \]

The given right side of the equation is:

\[ (3x + Ay) (3x^2 + Bxy + Cy^2) \]

Comparing the two sides:

\[ (3x - y)(3x^2 - 6xy + 7y^2) = (3x + Ay) (3x^2 + Bxy + Cy^2) \]

We can see that the terms in the factors correspond directly. By comparing the coefficients of the terms in the corresponding factors, we can find the values of A, B, and C.

Step 4: Compare Coefficients to Find A, B, and C

Compare the first factors: \((3x - y)\) and \((3x + Ay)\).

The coefficient of \(x\) is 3 on both sides, which matches.

The term with \(y\) is \(-y\) on the left and \(Ay\) on the right. Comparing the coefficients of \(y\), we get:

\[ -1 = A \]

So, \(A = -1\).

Compare the second factors: \((3x^2 - 6xy + 7y^2)\) and \((3x^2 + Bxy + Cy^2)\).

The coefficient of \(x^2\) is 3 on both sides, which matches.

The term with \(xy\) is \(-6xy\) on the left and \(Bxy\) on the right. Comparing the coefficients of \(xy\), we get:

\[ -6 = B \]

So, \(B = -6\).

The term with \(y^2\) is \(7y^2\) on the left and \(Cy^2\) on the right. Comparing the coefficients of \(y^2\), we get:

\[ 7 = C \]

So, \(C = 7\).

Step 5: Calculate the value of A + B + C

Now that we have the values for A, B, and C, we can find their sum:

\[ A + B + C = (-1) + (-6) + 7 \]

\[ = -1 - 6 + 7 \]

\[ = -7 + 7 = 0 \]

The value of \(A + B + C\) is 0.

Therefore, \(A + B + C = -1 + (-6) + 7 = 0\).

Revision Table: Key Algebraic Identities

Additional Information: Coefficient Comparison Method

The coefficient comparison method is a powerful technique used when two polynomial expressions are stated to be equal for all values of the variables. If two polynomials are equal, then the coefficients of corresponding terms (terms with the same variable powers, like \(x^2\), \(xy\), \(y^2\), etc.) must be equal.

In this problem, once we simplified the left side into a product of two factors and had it equal to the product on the right side, we could equate the corresponding factors directly:

Factor 1: \((3x - y) = (3x + Ay)\)

Factor 2: \((3x^2 - 6xy + 7y^2) = (3x^2 + Bxy + Cy^2)\)

By matching the coefficients of the corresponding terms within these equal factors, we could determine the unknown values A, B, and C. This method relies on the fundamental property that a polynomial is uniquely defined by its coefficients.