Understanding the Problem: Expanding and Simplifying Expressions

The question asks us to evaluate a specific expression involving the sum of two cubic terms and relate it to a given polynomial form. We are given the equation: \[ {{(4x + 2y)^3} + {(4x-2y)^3}}=16({Ax^3 + Bxy^2}) \] Our goal is to first simplify the left-hand side of this equation. Then, by comparing the simplified form with the right-hand side, we need to find the values of the constants \(A\) and \(B\). Finally, using these values of \(A\) and \(B\), we need to calculate the value of \( \frac{1}{2 }{(\sqrt{A^2 +B^2})} \).

Expanding the Cubic Terms using Algebraic Identities

To simplify the expression \( {(4x + 2y)^3} + {(4x-2y)^3} \), we can use the algebraic identities for the cube of a sum and the cube of a difference:

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

Let's apply these identities to our specific terms. Let \( a = 4x \) and \( b = 2y \). Then:

• \( (4x + 2y)^3 = (4x)^3 + 3(4x)^2(2y) + 3(4x)(2y)^2 + (2y)^3 \)
• \( (4x - 2y)^3 = (4x)^3 - 3(4x)^2(2y) + 3(4x)(2y)^2 - (2y)^3 \)

Now, let's simplify each expanded term:

• \( (4x)^3 = 64x^3 \)
• \( 3(4x)^2(2y) = 3(16x^2)(2y) = 96x^2y \)
• \( 3(4x)(2y)^2 = 3(4x)(4y^2) = 48xy^2 \)
• \( (2y)^3 = 8y^3 \)

So, we have:

• \( (4x + 2y)^3 = 64x^3 + 96x^2y + 48xy^2 + 8y^3 \)
• \( (4x - 2y)^3 = 64x^3 - 96x^2y + 48xy^2 - 8y^3 \)

Now, we add these two expanded forms together:

\[ {(4x + 2y)^3} + {(4x-2y)^3} = (64x^3 + 96x^2y + 48xy^2 + 8y^3) + (64x^3 - 96x^2y + 48xy^2 - 8y^3) \]

Combining like terms:

\[ {(4x + 2y)^3} + {(4x-2y)^3} = (64x^3 + 64x^3) + (96x^2y - 96x^2y) + (48xy^2 + 48xy^2) + (8y^3 - 8y^3) \]

\[ {(4x + 2y)^3} + {(4x-2y)^3} = 128x^3 + 0 + 96xy^2 + 0 \]

\[ {(4x + 2y)^3} + {(4x-2y)^3} = 128x^3 + 96xy^2 \]

Alternatively, we could use the identity \( (a+b)^3 + (a-b)^3 = 2a^3 + 6ab^2 \). Let \( a=4x \) and \( b=2y \). Then \( {(4x + 2y)^3} + {(4x-2y)^3} = 2(4x)^3 + 6(4x)(2y)^2 = 2(64x^3) + 6(4x)(4y^2) = 128x^3 + 24x(4y^2) = 128x^3 + 96xy^2 \). This gives the same simplified result.

Comparing Coefficients to Find A and B

We are given that \( {(4x + 2y)^3} + {(4x-2y)^3}}=16({Ax^3 + Bxy^2})\). From our simplification, we found that \( {(4x + 2y)^3} + {(4x-2y)^3} = 128x^3 + 96xy^2 \).

So, we have the equation:

\[ 128x^3 + 96xy^2 = 16(Ax^3 + Bxy^2) \]

Divide both sides by 16:

\[ \frac{128x^3 + 96xy^2}{16} = Ax^3 + Bxy^2 \]

\[ \frac{128x^3}{16} + \frac{96xy^2}{16} = Ax^3 + Bxy^2 \]

\[ 8x^3 + 6xy^2 = Ax^3 + Bxy^2 \]

By comparing the coefficients of the like terms (\(x^3\) and \(xy^2\)) on both sides of the equation, we can find the values of \(A\) and \(B\).

• Coefficient of \(x^3\): \( 8 = A \implies A = 8 \)
• Coefficient of \(xy^2\): \( 6 = B \implies B = 6 \)

So, the values are \(A = 8\) and \(B = 6\).

Calculating the Final Value

The question asks for the value of \( \frac{1}{2 }{(\sqrt{A^2 +B^2})} \). We have \( A = 8 \) and \( B = 6 \). Substitute these values into the expression:

\[ \frac{1}{2 }{(\sqrt{A^2 +B^2})} = \frac{1}{2 }{(\sqrt{8^2 +6^2})} \]

\[ = \frac{1}{2 }{(\sqrt{64 + 36})} \]

\[ = \frac{1}{2 }{(\sqrt{100})} \]

\[ = \frac{1}{2 }{(10)} \]

\[ = 5 \]

The value of \( \frac{1}{2 }{(\sqrt{A^2 +B^2})} \) is 5.

Revision Table: Key Concepts for Cubic Expansion and Equation Solving

Additional Information: Polynomial Identity Applications

The problem effectively uses polynomial identities and the principle of comparing coefficients. Understanding these concepts is crucial for solving many algebra problems.

• Polynomial Identity: An equation that is true for all possible values of the variables involved. The cubic expansion formulas \( (a+b)^3 \) and \( (a-b)^3 \) are examples of identities. The identity \( (a+b)^3 + (a-b)^3 = 2a^3 + 6ab^2 \) is derived from these basic ones and is very useful when dealing with sums of cubic terms like in this problem.
• Comparing Coefficients: If you have an equation where a polynomial on one side is equal to a polynomial on the other side, and this equation holds true for all values of the variables, then the coefficients of each term with the same power of the variables must be equal. This is a powerful technique for finding unknown constants within polynomial equations. For example, if \(cx^2 + dx + e = 5x^2 - 2x + 1\) for all \(x\), then by comparing coefficients, we know \(c=5\), \(d=-2\), and \(e=1\). In our problem, we compared coefficients of \(x^3\) and \(xy^2\) to find \(A\) and \(B\).
• Applications: These techniques are widely used in algebra, calculus (e.g., partial fraction decomposition), and various fields of science and engineering where polynomial equations are involved.