Algebraic Identity: Simplifying Cubic Expressions

The problem asks us to find the value of the expression \((C - A + B)\) given an algebraic identity. The identity provided is:

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

Understanding the Sum of Cubes Identity

This algebraic expression can be simplified using the sum of cubes formula, which states:

\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)

In our given equation, let's identify \(a\) and \(b\):

• Let \(a = (x + y)\)
• We have \(8(x - y)^3\), which can be written as \([2(x - y)]^3\). So, let \(b = 2(x - y)\).

Now, we will apply the sum of cubes formula to the left-hand side of the given equation.

Step-by-Step Expansion and Comparison

1. Expanding the Term \((a + b)\)

First, calculate the sum of \(a\) and \(b\):

\(a + b = (x + y) + 2(x - y)\)

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

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

\(= 3x - y\)

Comparing this with the first factor on the right-hand side, \((3x + Ay)\):

\(3x - y = 3x + Ay\)

By comparing the coefficient of \(y\), we find:

\(A = -1\)

2. Expanding the Term \((a^2 - ab + b^2)\)

Next, we need to calculate \(a^2\), \(b^2\), and \(ab\).

• Calculate \(a^2\):
  \(a^2 = (x + y)^2 = x^2 + 2xy + y^2\)
• Calculate \(b^2\):
  \(b^2 = [2(x - y)]^2 = 4(x - y)^2 = 4(x^2 - 2xy + y^2) = 4x^2 - 8xy + 4y^2\)
• Calculate \(ab\):
  \(ab = (x + y)[2(x - y)] = 2(x + y)(x - y) = 2(x^2 - y^2) = 2x^2 - 2y^2\)

Now, substitute these into the expression \((a^2 - ab + b^2)\):

\(a^2 - ab + b^2 = (x^2 + 2xy + y^2) - (2x^2 - 2y^2) + (4x^2 - 8xy + 4y^2)\)

Carefully distribute the negative sign and combine like terms:

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

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

So, the expanded form is \((a^2 - ab + b^2) = 3x^2 - 6xy + 7y^2\).

Comparing this with the second factor on the right-hand side, \((3x^2 + Bxy + Cy^2)\):

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

By comparing the coefficients of \(xy\) and \(y^2\), we find:

• \(B = -6\)
• \(C = 7\)

Calculating the Value of \((C - A + B)\)

We have found the values of A, B, and C:

• \(A = -1\)
• \(B = -6\)
• \(C = 7\)

Now, substitute these values into the expression \((C - A + B)\):

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

\(= 7 + 1 - 6\)

\(= 8 - 6\)

\(= 2\)

Therefore, the value of \((C - A + B)\) is 2.