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### **Activity 3: Whole to Parts**

The task requires selecting the correct factors for the given polynomials from the provided options. Let's go through each polynomial and identify its factors:

1. **\(x^2 - 81\)**:
   - This is a difference of squares: \(x^2 - 81 = (x - 9)(x + 9)\).
   - **Answer: (x - 9)(x + 9)**

2. **\(x^2 + 4x + 4\)**:
   - This is a perfect square trinomial: \(x^2 + 4x + 4 = (x + 2)^2\).
   - **Answer: (x + 2)^2**

3. **\(16x^2 - 81\)**:
   - This is another difference of squares: \(16x^2 - 81 = (4x - 9)(4x + 9)\).
   - **Answer: (4x - 9)(4x + 9)**

4. **\(9x^2 - 12xy + 4y^2\)**:
   - This is a perfect square trinomial: \(9x^2 - 12xy + 4y^2 = (3x - 2y)^2\).
   - **Answer: (3x - 2y)^2**

5. **\(4x^2 - 25y^2\)**:
   - This is a difference of squares: \(4x^2 - 25y^2 = (2x - 5y)(2x + 5y)\).
   - **Answer: (2x - 5y)(2x + 5y)**

### **Activity 4: When I Broke It, I Found It!**

The task requires finding the missing terms to factor the polynomials completely.

1. **\(3x^2 - 12x^7 + 9x^4\)**
   - Factor out the greatest common factor, \(3x^2\):
   - \(3x^2(x^0 - 4x^5 + 3x^2)\).
   - The missing term is \(3x^2\), so the expression is:
   - **Final Factors: \(3x^2(1 - 4x^5 + 3x^2)\)**

2. **\(x^4 - 16x^4 + x^2 + x^2\)**
   - Simplify and combine like terms:
   - \(x^4 - 16x^4 + 2x^2 = (x^2(1 - 16x^2) + 2x^2)\).
   - The missing term is \(x^2\), so the expression is:
   - **Final Factors: \(x^2(1 - 16x^2 + 2)\)**

3. **\(4x^4 - 625x^2 + 25x^4 = (x^2 + 5)\)**
   - Factor out \(4x^4 + 25x^4 - 625x^2\) to find:
   - \(x^4(4x^2 + 25 - 625) = 4x^2(25 + 25)\).
   - The missing term is \(25\), so the expression is:
   - **Final Factors: \(25(4x^2 - 625x^2)\)**

Would you like further details or clarifications? If so, feel free to ask.

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Here are some follow-up questions:

1. What is the general formula for factoring a difference of squares?
2. Can you factor \(x^2 - 4x + 4\) as a perfect square trinomial?
3. How would you factor a polynomial with three terms that is not a perfect square trinomial?
4. Can you identify any polynomial that is neither a difference of squares nor a perfect square trinomial?
5. How can you verify your factors by expanding the products?
6. How does factoring help in solving quadratic equations?
7. What are common mistakes when factoring complex polynomials?
8. How would you factor \(x^2 + 4x - 5\)?

**Tip:** When factoring polynomials, always check for a greatest common factor (GCF) before trying other methods like difference of squares or perfect square trinomials.

Learn how to solve polynomial factoring problems with detailed step-by-step solutions. This includes factoring techniques like difference of squares and perfect square trinomials. Suitable for students in Grades 9-12.

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