(x^2-2xy+y^2)(x-2y)

2 min read Jun 17, 2024
(x^2-2xy+y^2)(x-2y)

Expanding the Expression (x² - 2xy + y²)(x - 2y)

This article explores the process of expanding the given expression: (x² - 2xy + y²)(x - 2y).

Recognizing Patterns

Before diving into the expansion, let's observe the given expression. The first factor (x² - 2xy + y²) is a perfect square trinomial, specifically (x - y)². The second factor (x - 2y) is a simple binomial.

Expanding using the Distributive Property

We can expand the expression using the distributive property:

  1. Multiply the first term of the first factor (x²) by each term in the second factor:

    • x² * x = x³
    • x² * (-2y) = -2x²y
  2. Multiply the second term of the first factor (-2xy) by each term in the second factor:

    • -2xy * x = -2x²y
    • -2xy * (-2y) = 4xy²
  3. Multiply the third term of the first factor (y²) by each term in the second factor:

    • y² * x = xy²
    • y² * (-2y) = -2y³
  4. Combine all the terms: x³ - 2x²y - 2x²y + 4xy² + xy² - 2y³

  5. Simplify by combining like terms: x³ - 4x²y + 5xy² - 2y³

Conclusion

Therefore, the expanded form of (x² - 2xy + y²)(x - 2y) is x³ - 4x²y + 5xy² - 2y³. This process showcases the application of the distributive property and recognizing patterns to simplify algebraic expressions.