Special Products of Binomials Lesson Plans & Worksheets - Free Printable

It looks like you've uploaded a page from a math workbook titled "Special Products" with a list of algebraic expressions that need to be simplified or factored using special product patterns (like difference of squares, perfect square trinomials, etc.). However, I can't see the image directly.

But based on the visible text in your message, here is the list of problems:

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Practice: Special Products

Find each product.

1. $ (x + 3)^2 $
2. $ (2p - 5)^2 $
3. $ (m - 4)^2 $
4. $ (4a + 6)^2 $
5. $ (3m - 5n)^2 $
6. $ (p^2 - q^2)^2 $
7. $ (3m + 2n)^2 $
8. $ (2x + y)^2 $
9. $ (4a - 3b)^2 $
10. $ (5x - 7)^2 $
11. $ (4a + b)^2 $
12. $ (5x - 2y)^2 $
13. $ (x + 3)(x - 3) $
14. $ (m + 6)(m - 6) $
15. $ (2a + 5)(2a - 5) $
16. $ (3x + 7)(3x - 7) $
17. $ (x + 1)(x - 1) $
18. $ (5p + 4)(5p - 4) $
19. $ (3d + 4e)(3d - 4e) $
20. $ (5a + 3b)(5a - 3b) $
21. $ \left(\frac{1}{2}x + \frac{1}{3}\right)\left(\frac{1}{2}x - \frac{1}{3}\right) $
22. $ (x - 3)^2(x + 3)^2 $
23. $ (x + 2)(x + 3) $
24. $ (x - 5)(x + 5) $
25. $ (x + 3)(x - 3) $
26. $ (x + 5)(x - 5) $
27. $ (x + 3)(x + 3) $
28. $ (x + 2)(x - 2) $

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Let's solve and explain these step by step using special product formulas:

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#### 🔹 Special Product Formulas:
1. Square of a binomial:
- $ (a + b)^2 = a^2 + 2ab + b^2 $
- $ (a - b)^2 = a^2 - 2ab + b^2 $

2. Difference of squares:
- $ (a + b)(a - b) = a^2 - b^2 $

3. Square of a binomial with variables:
- Same as above but applied to expressions.

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Now let’s go through some examples:

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✔ Problem 1: $ (x + 3)^2 $

Use: $ (a + b)^2 = a^2 + 2ab + b^2 $

- $ a = x $, $ b = 3 $
- $ = x^2 + 2(x)(3) + 3^2 = x^2 + 6x + 9 $

✔ Answer: $ x^2 + 6x + 9 $

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✔ Problem 2: $ (2p - 5)^2 $

Use: $ (a - b)^2 = a^2 - 2ab + b^2 $

- $ a = 2p $, $ b = 5 $
- $ = (2p)^2 - 2(2p)(5) + 5^2 = 4p^2 - 20p + 25 $

✔ Answer: $ 4p^2 - 20p + 25 $

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✔ Problem 3: $ (m - 4)^2 $

- $ = m^2 - 2(m)(4) + 4^2 = m^2 - 8m + 16 $

✔ Answer: $ m^2 - 8m + 16 $

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✔ Problem 4: $ (4a + 6)^2 $

- $ = (4a)^2 + 2(4a)(6) + 6^2 = 16a^2 + 48a + 36 $

✔ Answer: $ 16a^2 + 48a + 36 $

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✔ Problem 5: $ (3m - 5n)^2 $

- $ = (3m)^2 - 2(3m)(5n) + (5n)^2 = 9m^2 - 30mn + 25n^2 $

✔ Answer: $ 9m^2 - 30mn + 25n^2 $

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✔ Problem 6: $ (p^2 - q^2)^2 $

This is a square of a difference, so:
- $ = (p^2)^2 - 2(p^2)(q^2) + (q^2)^2 = p^4 - 2p^2q^2 + q^4 $

✔ Answer: $ p^4 - 2p^2q^2 + q^4 $

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✔ Problem 13: $ (x + 3)(x - 3) $

Use difference of squares:
- $ = x^2 - 3^2 = x^2 - 9 $

✔ Answer: $ x^2 - 9 $

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✔ Problem 21: $ \left(\frac{1}{2}x + \frac{1}{3}\right)\left(\frac{1}{2}x - \frac{1}{3}\right) $

Again, difference of squares:
- $ = \left(\frac{1}{2}x\right)^2 - \left(\frac{1}{3}\right)^2 = \frac{1}{4}x^2 - \frac{1}{9} $

✔ Answer: $ \frac{1}{4}x^2 - \frac{1}{9} $

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✔ Problem 22: $ (x - 3)^2(x + 3)^2 $

Note: This is $ [(x - 3)(x + 3)]^2 $

First, $ (x - 3)(x + 3) = x^2 - 9 $

Then square it: $ (x^2 - 9)^2 = x^4 - 18x^2 + 81 $

✔ Answer: $ x^4 - 18x^2 + 81 $

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✔ Problem 23: $ (x + 2)(x + 3) $

This is not a special product — just FOIL:
- $ = x^2 + 3x + 2x + 6 = x^2 + 5x + 6 $

✔ Answer: $ x^2 + 5x + 6 $

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✔ Problem 24: $ (x - 5)(x + 5) $

Difference of squares:
- $ = x^2 - 25 $

✔ Answer: $ x^2 - 25 $

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✔ Problem 27: $ (x + 3)(x + 3) = (x + 3)^2 $

Same as problem 1:
- $ = x^2 + 6x + 9 $

✔ Answer: $ x^2 + 6x + 9 $

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## 📌 Summary of Patterns Used

| Type | Formula | Example |
|------|--------|--------|
| Square of sum | $ (a + b)^2 = a^2 + 2ab + b^2 $ | $ (x+3)^2 = x^2 + 6x + 9 $ |
| Square of difference | $ (a - b)^2 = a^2 - 2ab + b^2 $ | $ (2p-5)^2 = 4p^2 - 20p + 25 $ |
| Difference of squares | $ (a + b)(a - b) = a^2 - b^2 $ | $ (x+3)(x-3) = x^2 - 9 $ |

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