This problem involves simplifying a complex algebraic expression using given definitions for the variables \(p\), \(q\), and \(r\).

We are given:

• \(p = x^{4} - y^{2}z^{2}\)
• \(q = y^{4} - z^{2}x^{2}\)
• \(r = z^{4} - x^{2}y^{2}\)

We need to find the value of the expression: \(x^{2} (px^{2} + qy^{2} + rz^{2}) + qr - p^{2}\).

Step-by-Step Expression Simplification

Let's break down the problem into smaller parts and evaluate each term separately before combining them.

Step 1: Evaluate the term \(x^{2} (px^{2} + qy^{2} + rz^{2})\)

Substitute the expressions for \(p\), \(q\), and \(r\) into this part:

\(x^{2} ( (x^{4} - y^{2}z^{2})x^{2} + (y^{4} - z^{2}x^{2})y^{2} + (z^{4} - x^{2}y^{2})z^{2} )\)

Distribute the terms inside the parenthesis:

\(x^{2} ( x^{6} - x^{2}y^{2}z^{2} + y^{6} - z^{2}x^{2}y^{2} + z^{6} - x^{2}y^{2}z^{2} )\)

Combine like terms inside the parenthesis:

\(x^{2} ( x^{6} + y^{6} + z^{6} - 3x^{2}y^{2}z^{2} )\)

Now distribute the \(x^{2}\) outside:

\(x^{8} + x^{2}y^{6} + x^{2}z^{6} - 3x^{4}y^{2}z^{2}\)

Step 2: Evaluate the term \(qr\)

Multiply the expressions for \(q\) and \(r\):

\(qr = (y^{4} - z^{2}x^{2})(z^{4} - x^{2}y^{2})\)

Expand the product:

\(qr = y^{4}z^{4} - y^{4}x^{2}y^{2} - z^{2}x^{2}z^{4} + z^{2}x^{2}x^{2}y^{2}\)

Simplify the terms:

\(qr = y^{4}z^{4} - x^{2}y^{6} - x^{2}z^{6} + x^{4}y^{2}z^{2}\)

Step 3: Evaluate the term \(p^{2}\)

Square the expression for \(p\):

\(p^{2} = (x^{4} - y^{2}z^{2})^{2}\)

Using the algebraic identity \((a-b)^{2} = a^{2} - 2ab + b^{2}\):

\(p^{2} = (x^{4})^{2} - 2(x^{4})(y^{2}z^{2}) + (y^{2}z^{2})^{2}\)

Simplify the terms:

\(p^{2} = x^{8} - 2x^{4}y^{2}z^{2} + y^{4}z^{4}\)

Step 4: Combine the evaluated terms

Now substitute the results from Steps 1, 2, and 3 into the original expression: \(x^{2} (px^{2} + qy^{2} + rz^{2}) + qr - p^{2}\)

Original Expression = (Result from Step 1) + (Result from Step 2) - (Result from Step 3)

Original Expression = \((x^{8} + x^{2}y^{6} + x^{2}z^{6} - 3x^{4}y^{2}z^{2}) + (y^{4}z^{4} - x^{2}y^{6} - x^{2}z^{6} + x^{4}y^{2}z^{2}) - (x^{8} - 2x^{4}y^{2}z^{2} + y^{4}z^{4})\)

Remove the parentheses. Remember to change the signs of the terms inside the last parenthesis because of the minus sign before it:

\(x^{8} + x^{2}y^{6} + x^{2}z^{6} - 3x^{4}y^{2}z^{2} + y^{4}z^{4} - x^{2}y^{6} - x^{2}z^{6} + x^{4}y^{2}z^{2} - x^{8} + 2x^{4}y^{2}z^{2} - y^{4}z^{4}\)

Group and cancel out terms:

• The \(x^{8}\) term cancels with \(-x^{8}\).
• The \(x^{2}y^{6}\) term cancels with \(-x^{2}y^{6}\).
• The \(x^{2}z^{6}\) term cancels with \(-x^{2}z^{6}\).
• The \(y^{4}z^{4}\) term cancels with \(-y^{4}z^{4}\).

After cancellation, the expression becomes:

\(-3x^{4}y^{2}z^{2} + x^{4}y^{2}z^{2} + 2x^{4}y^{2}z^{2}\)

Combine the remaining terms:

\((-3 + 1 + 2)x^{4}y^{2}z^{2} = (0)x^{4}y^{2}z^{2} = 0\)

Final Result

The value of the expression \(x^{2} (px^{2} + qy^{2} + rz^{2}) + qr - p^{2}\) is \(0\).

Revision Table: Key Concepts

Additional Information on Algebraic Simplification

Algebraic simplification is a fundamental skill in mathematics. It involves rewriting an algebraic expression in a simpler form. This can make expressions easier to understand, evaluate, or manipulate in further calculations.

• Order of Operations: When simplifying, follow the standard order of operations (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), often remembered by acronyms like PEMDAS or BODMAS.
• Distributive Property: This property, \(a(b+c) = ab + ac\), is key for expanding expressions like \(x^{2}(px^{2} + qy^{2} + rz^{2})\).
• Factoring: The reverse of expansion, factoring involves writing an expression as a product of its factors. While not the primary method used here, it's another important simplification technique.
• Recognizing Patterns: Being familiar with common algebraic identities (\((a+b)^{2}\), \((a-b)^{2}\), \((a+b)(a-b)\), etc.) can significantly speed up simplification.

In this specific problem, careful application of substitution, expansion, and combining like terms led directly to the simplified result of \(0\).