Question: Expand the product $ (x + 2y - z)(x - 2y + z) $ and simplify the result.
["SEO-Optimized Article: Expand and Simplify the Product $ (x + 2y - z)(x - 2y + z) $", "Understanding how to expand and simplify algebraic expressions is fundamental in algebra and prepares students and learners for more advanced problem-solving. One commonly encountered expression is $ (x + 2y - z)(x - 2y + z) $. In this article, we explore how to expand this product step-by-step and simplify the result efficiently.", "---", "### What is the Expression?", "We start with the expression:\n$$\n(x + 2y - z)(x - 2y + z)\n$$", "At first glance, this looks like a product involving binomials with both positive and negative signs. This form resembles the identity:\n$$\n(a + b)(a - b) = a^2 - b^2\n$$\nHowever, our expression is slightly more complex, combining both $ 2y $ and $ z $ with alternating signs.", "---", "### Step 1: Introduction of Substitution for Clarity", "To simplify manipulations, consider grouping terms smartly. Let us define:\n- $ a = x $\n- $ b = 2y - z $", "Then the expression becomes:\n$$\n(x + b)(x - b + 2z) \quad \ ext{— wait, this doesn’t help directly.}\n$$", "A better approach is to treat $ (x + (2y - z))(x - (2y - z)) $, which clearly fits the difference of squares pattern:\n$$\n(a + b)(a - b) = a^2 - b^2\n$$\nHere:\n- $ a = x $\n- $ b = 2y - z $", "So:\n$$\n(x + 2y - z)(x - 2y + z) = \left(x + (2y - z)\right)\left(x - (2y - z)\right) = x^2 - (2y - z)^2\n$$", "---", "### Step 2: Expand $ (2y - z)^2 $", "Now simplify the right-hand side:\n$$\nx^2 - (2y - z)^2\n$$", "Expand $ (2y - z)^2 $:\n$$\n(2y - z)^2 = (2y)^2 - 2(2y)(z) + z^2 = 4y^2 - 4yz + z^2\n$$", "So the full expression becomes:\n$$\nx^2 - (4y^2 - 4yz + z^2) = x^2 - 4y^2 + 4yz - z^2\n$$", "---", "### Final Simplified Form", "After combining all terms, the expanded and simplified result is:\n$$\nx^2 - 4y^2 + 4yz - z^2\n$$", "---", "### Why This Matters: Benefits of Expanding and Simplifying", "- Reduces complexity: Turning a product into a simplified sum improves clarity.\n- Reveals structure: Recognizing patterns like the difference of squares streamlines computation.\n- Facilitates further calculation: Useful in calculus, physics, and engineering for evaluating expressions and solving equations.", "---", "### Conclusion", "To expand $ (x + 2y - z)(x - 2y + z) $, recognize it as a difference of squares:\n$$\n(x + (2y - z))(x - (2y - z)) = x^2 - (2y - z)^2\n$$\nThen fully expand and combine like terms to reach the simplified form:\n$$\n\boxed{x^2 - 4y^2 + 4yz - z^2}\n$$", "Mastering such algebraic manipulations strengthens your mathematical foundation and enhances problem-solving skills across STEM fields.", "---", "### Key SEO Keywords:\nexpand $ (x + 2y - z)(x - 2y + z) $, simplify product, difference of squares expansion, algebraic identities, step-by-step expansion", "---", "Headings used for SEO:\n- What is $ (x + 2y - z)(x - 2y + z) $?\n- Expand and Simplify: $ (x + 2y - z)(x - 2y + z) $\n- Step-by-step: Difference of Squares in Algebra\n- Final Simplified Form: $ x^2 - 4y^2 + 4yz - z^2 $", "This structured, keyword-rich article supports learning, improves SEO ranking, and helps students master algebraic expansion effectively."]
