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(x^2+y^2-5)^2-4x^2y^2-16xy-16

3 min read Jun 17, 2024
(x^2+y^2-5)^2-4x^2y^2-16xy-16

Factoring the Expression (x^2 + y^2 - 5)^2 - 4x^2y^2 - 16xy - 16

This expression appears complex, but we can simplify it through careful factoring. Here's a step-by-step breakdown:

1. Recognizing Patterns

  • Difference of Squares: Notice that the first term is a perfect square: (x^2 + y^2 - 5)^2.
  • Perfect Square Trinomial: The remaining terms (-4x^2y^2 - 16xy - 16) can be rearranged and factored into a perfect square trinomial.

2. Applying the Patterns

  • Difference of Squares: (a^2 - b^2) = (a + b)(a - b)
  • Perfect Square Trinomial: (a^2 + 2ab + b^2) = (a + b)^2

Let's apply these:

  1. Rewrite the expression: (x^2 + y^2 - 5)^2 - (2xy + 4)^2

  2. Factor using the difference of squares: [(x^2 + y^2 - 5) + (2xy + 4)][(x^2 + y^2 - 5) - (2xy + 4)]

  3. Simplify: (x^2 + y^2 + 2xy - 1)(x^2 + y^2 - 2xy - 9)

  4. Factor the remaining terms: [(x + y)^2 - 1][(x - y)^2 - 9]

  5. Apply difference of squares again: [(x + y + 1)(x + y - 1)][(x - y + 3)(x - y - 3)]

Final Result

The factored form of the expression (x^2 + y^2 - 5)^2 - 4x^2y^2 - 16xy - 16 is:

(x + y + 1)(x + y - 1)(x - y + 3)(x - y - 3)

Key Takeaways

  • Recognize patterns: Identifying patterns like the difference of squares and perfect square trinomials is crucial for simplifying expressions.
  • Break it down: Factor the expression step by step, using the appropriate pattern at each stage.
  • Simplify and refine: Once factored, check for further simplification opportunities.

This exercise demonstrates how factoring can transform complex expressions into more manageable and understandable forms.

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