Special Products of Binomials Lesson Plans & Worksheets - Free Printable
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Step-by-step solution for: Special Products of Binomials Lesson Plans & Worksheets
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Show Answer Key & Explanations
Step-by-step solution for: Special Products of Binomials Lesson Plans & Worksheets
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. The goal is likely to apply special product formulas such as:
- Difference of squares: $ a^2 - b^2 = (a - b)(a + b) $
- Perfect square trinomials:
$ a^2 + 2ab + b^2 = (a + b)^2 $
$ a^2 - 2ab + b^2 = (a - b)^2 $
- Sum and difference of cubes:
$ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $
$ a^3 - b^3 = (a - b)(a^2 + ab + b^2) $
Let’s go through the problems one by one and solve them using these patterns.
---
1. $ 9x^2 - 4y^2 $
→ This is a difference of squares:
$ = (3x)^2 - (2y)^2 = (3x - 2y)(3x + 2y) $
2. $ 4x^2 - 81 $
→ $ = (2x)^2 - 9^2 = (2x - 9)(2x + 9) $
3. $ 36 - 121 $
→ Wait — this is just numbers:
$ 36 - 121 = -85 $
But if it's meant to be $ 36x^2 - 121 $, then:
$ = (6x)^2 - 11^2 = (6x - 11)(6x + 11) $
But as written: $ 36 - 121 = -85 $ — likely a typo.
4. $ 49 - p^2 $
→ $ = 7^2 - p^2 = (7 - p)(7 + p) $
5. $ 100 - 25x^2 $
→ Factor out GCF first:
$ = 25(4 - x^2) = 25(2 - x)(2 + x) $
6. $ y^2 - 16x^2 $
→ $ = y^2 - (4x)^2 = (y - 4x)(y + 4x) $
7. $ 25m^2 - p^2 $
→ $ = (5m)^2 - p^2 = (5m - p)(5m + p) $
8. $ 16x^2 - 9 $
→ $ = (4x)^2 - 3^2 = (4x - 3)(4x + 3) $
9. $ 16x^2 - 9 $ again? Wait, same as #8? Possibly duplicate.
Wait — let's check the actual list more carefully.
Looking at your image, here are the problems (numbered 1–26):
1. $ 9x^2 - 4y^2 $
2. $ 4x^2 - 81 $
3. $ 36 - 121 $ ← likely $ 36x^2 - 121 $?
4. $ 49 - p^2 $
5. $ 100 - 25x^2 $
6. $ y^2 - 16x^2 $
7. $ 25m^2 - p^2 $
8. $ 16x^2 - 9 $
9. $ 36x^2 - 25 $
10. $ 25x^2 - 49 $
11. $ 4x^2 - 9y^2 $
12. $ 25x^2 - 16y^2 $
13. $ 16x^2 - 25y^2 $
14. $ 49x^2 - 16y^2 $
15. $ 121x^2 - 100y^2 $
16. $ 4x^2 - 81y^2 $
17. $ (2x + 1)(2x - 1) $ ← already factored?
18. $ (3x + 4)(3x - 4) $
19. $ (4x + 5)(4x - 5) $
20. $ (5x + 7)(5x - 7) $
21. $ (6x + 8)(6x - 8) $
22. $ (7x + 9)(7x - 9) $
23. $ (8x + 10)(8x - 10) $
24. $ (9x + 11)(9x - 11) $
25. $ (10x + 12)(10x - 12) $
26. $ (11x + 13)(11x - 13) $
Wait — actually, looking closely, problems 17–26 seem to be already in factored form, but they may be asking to expand them?
But earlier ones are unfactored, so likely:
- Problems 1–16: Factor the expression
- Problems 17–26: Expand the product
Let’s confirm.
---
We'll use difference of squares pattern: $ a^2 - b^2 = (a - b)(a + b) $
1. $ 9x^2 - 4y^2 = (3x)^2 - (2y)^2 = (3x - 2y)(3x + 2y) $
2. $ 4x^2 - 81 = (2x)^2 - 9^2 = (2x - 9)(2x + 9) $
3. $ 36 - 121 = -85 $ — but probably meant $ 36x^2 - 121 $ →
$ = (6x)^2 - 11^2 = (6x - 11)(6x + 11) $
4. $ 49 - p^2 = 7^2 - p^2 = (7 - p)(7 + p) $
5. $ 100 - 25x^2 = 25(4 - x^2) = 25(2 - x)(2 + x) $
6. $ y^2 - 16x^2 = y^2 - (4x)^2 = (y - 4x)(y + 4x) $
7. $ 25m^2 - p^2 = (5m)^2 - p^2 = (5m - p)(5m + p) $
8. $ 16x^2 - 9 = (4x)^2 - 3^2 = (4x - 3)(4x + 3) $
9. $ 36x^2 - 25 = (6x)^2 - 5^2 = (6x - 5)(6x + 5) $
10. $ 25x^2 - 49 = (5x)^2 - 7^2 = (5x - 7)(5x + 7) $
11. $ 4x^2 - 9y^2 = (2x)^2 - (3y)^2 = (2x - 3y)(2x + 3y) $
12. $ 25x^2 - 16y^2 = (5x)^2 - (4y)^2 = (5x - 4y)(5x + 4y) $
13. $ 16x^2 - 25y^2 = (4x)^2 - (5y)^2 = (4x - 5y)(4x + 5y) $
14. $ 49x^2 - 16y^2 = (7x)^2 - (4y)^2 = (7x - 4y)(7x + 4y) $
15. $ 121x^2 - 100y^2 = (11x)^2 - (10y)^2 = (11x - 10y)(11x + 10y) $
16. $ 4x^2 - 81y^2 = (2x)^2 - (9y)^2 = (2x - 9y)(2x + 9y) $
---
Use identity: $ (a + b)(a - b) = a^2 - b^2 $
17. $ (2x + 1)(2x - 1) = (2x)^2 - (1)^2 = 4x^2 - 1 $
18. $ (3x + 4)(3x - 4) = (3x)^2 - 4^2 = 9x^2 - 16 $
19. $ (4x + 5)(4x - 5) = (4x)^2 - 5^2 = 16x^2 - 25 $
20. $ (5x + 7)(5x - 7) = 25x^2 - 49 $
21. $ (6x + 8)(6x - 8) = 36x^2 - 64 $
22. $ (7x + 9)(7x - 9) = 49x^2 - 81 $
23. $ (8x + 10)(8x - 10) = 64x^2 - 100 $
24. $ (9x + 11)(9x - 11) = 81x^2 - 121 $
25. $ (10x + 12)(10x - 12) = 100x^2 - 144 $
26. $ (11x + 13)(11x - 13) = 121x^2 - 169 $
---
#### Factoring (1–16):
1. $ (3x - 2y)(3x + 2y) $
2. $ (2x - 9)(2x + 9) $
3. $ (6x - 11)(6x + 11) $ *(assuming typo)*
4. $ (7 - p)(7 + p) $
5. $ 25(2 - x)(2 + x) $
6. $ (y - 4x)(y + 4x) $
7. $ (5m - p)(5m + p) $
8. $ (4x - 3)(4x + 3) $
9. $ (6x - 5)(6x + 5) $
10. $ (5x - 7)(5x + 7) $
11. $ (2x - 3y)(2x + 3y) $
12. $ (5x - 4y)(5x + 4y) $
13. $ (4x - 5y)(4x + 5y) $
14. $ (7x - 4y)(7x + 4y) $
15. $ (11x - 10y)(11x + 10y) $
16. $ (2x - 9y)(2x + 9y) $
#### Expanding (17–26):
17. $ 4x^2 - 1 $
18. $ 9x^2 - 16 $
19. $ 16x^2 - 25 $
20. $ 25x^2 - 49 $
21. $ 36x^2 - 64 $
22. $ 49x^2 - 81 $
23. $ 64x^2 - 100 $
24. $ 81x^2 - 121 $
25. $ 100x^2 - 144 $
26. $ 121x^2 - 169 $
---
All problems involve special products, specifically the difference of squares pattern:
$$
a^2 - b^2 = (a - b)(a + b)
$$
This is used both for factoring binomials and expanding products of conjugates.
The key is recognizing when an expression fits the form $ a^2 - b^2 $, and applying the formula accordingly.
Let me know if you'd like this formatted in a table or want to see any specific step in detail!
- Difference of squares: $ a^2 - b^2 = (a - b)(a + b) $
- Perfect square trinomials:
$ a^2 + 2ab + b^2 = (a + b)^2 $
$ a^2 - 2ab + b^2 = (a - b)^2 $
- Sum and difference of cubes:
$ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $
$ a^3 - b^3 = (a - b)(a^2 + ab + b^2) $
Let’s go through the problems one by one and solve them using these patterns.
---
Problems and Solutions
1. $ 9x^2 - 4y^2 $
→ This is a difference of squares:
$ = (3x)^2 - (2y)^2 = (3x - 2y)(3x + 2y) $
2. $ 4x^2 - 81 $
→ $ = (2x)^2 - 9^2 = (2x - 9)(2x + 9) $
3. $ 36 - 121 $
→ Wait — this is just numbers:
$ 36 - 121 = -85 $
But if it's meant to be $ 36x^2 - 121 $, then:
$ = (6x)^2 - 11^2 = (6x - 11)(6x + 11) $
But as written: $ 36 - 121 = -85 $ — likely a typo.
4. $ 49 - p^2 $
→ $ = 7^2 - p^2 = (7 - p)(7 + p) $
5. $ 100 - 25x^2 $
→ Factor out GCF first:
$ = 25(4 - x^2) = 25(2 - x)(2 + x) $
6. $ y^2 - 16x^2 $
→ $ = y^2 - (4x)^2 = (y - 4x)(y + 4x) $
7. $ 25m^2 - p^2 $
→ $ = (5m)^2 - p^2 = (5m - p)(5m + p) $
8. $ 16x^2 - 9 $
→ $ = (4x)^2 - 3^2 = (4x - 3)(4x + 3) $
9. $ 16x^2 - 9 $ again? Wait, same as #8? Possibly duplicate.
Wait — let's check the actual list more carefully.
Looking at your image, here are the problems (numbered 1–26):
1. $ 9x^2 - 4y^2 $
2. $ 4x^2 - 81 $
3. $ 36 - 121 $ ← likely $ 36x^2 - 121 $?
4. $ 49 - p^2 $
5. $ 100 - 25x^2 $
6. $ y^2 - 16x^2 $
7. $ 25m^2 - p^2 $
8. $ 16x^2 - 9 $
9. $ 36x^2 - 25 $
10. $ 25x^2 - 49 $
11. $ 4x^2 - 9y^2 $
12. $ 25x^2 - 16y^2 $
13. $ 16x^2 - 25y^2 $
14. $ 49x^2 - 16y^2 $
15. $ 121x^2 - 100y^2 $
16. $ 4x^2 - 81y^2 $
17. $ (2x + 1)(2x - 1) $ ← already factored?
18. $ (3x + 4)(3x - 4) $
19. $ (4x + 5)(4x - 5) $
20. $ (5x + 7)(5x - 7) $
21. $ (6x + 8)(6x - 8) $
22. $ (7x + 9)(7x - 9) $
23. $ (8x + 10)(8x - 10) $
24. $ (9x + 11)(9x - 11) $
25. $ (10x + 12)(10x - 12) $
26. $ (11x + 13)(11x - 13) $
Wait — actually, looking closely, problems 17–26 seem to be already in factored form, but they may be asking to expand them?
But earlier ones are unfactored, so likely:
- Problems 1–16: Factor the expression
- Problems 17–26: Expand the product
Let’s confirm.
---
Part A: Factoring (Problems 1–16)
We'll use difference of squares pattern: $ a^2 - b^2 = (a - b)(a + b) $
1. $ 9x^2 - 4y^2 = (3x)^2 - (2y)^2 = (3x - 2y)(3x + 2y) $
2. $ 4x^2 - 81 = (2x)^2 - 9^2 = (2x - 9)(2x + 9) $
3. $ 36 - 121 = -85 $ — but probably meant $ 36x^2 - 121 $ →
$ = (6x)^2 - 11^2 = (6x - 11)(6x + 11) $
4. $ 49 - p^2 = 7^2 - p^2 = (7 - p)(7 + p) $
5. $ 100 - 25x^2 = 25(4 - x^2) = 25(2 - x)(2 + x) $
6. $ y^2 - 16x^2 = y^2 - (4x)^2 = (y - 4x)(y + 4x) $
7. $ 25m^2 - p^2 = (5m)^2 - p^2 = (5m - p)(5m + p) $
8. $ 16x^2 - 9 = (4x)^2 - 3^2 = (4x - 3)(4x + 3) $
9. $ 36x^2 - 25 = (6x)^2 - 5^2 = (6x - 5)(6x + 5) $
10. $ 25x^2 - 49 = (5x)^2 - 7^2 = (5x - 7)(5x + 7) $
11. $ 4x^2 - 9y^2 = (2x)^2 - (3y)^2 = (2x - 3y)(2x + 3y) $
12. $ 25x^2 - 16y^2 = (5x)^2 - (4y)^2 = (5x - 4y)(5x + 4y) $
13. $ 16x^2 - 25y^2 = (4x)^2 - (5y)^2 = (4x - 5y)(4x + 5y) $
14. $ 49x^2 - 16y^2 = (7x)^2 - (4y)^2 = (7x - 4y)(7x + 4y) $
15. $ 121x^2 - 100y^2 = (11x)^2 - (10y)^2 = (11x - 10y)(11x + 10y) $
16. $ 4x^2 - 81y^2 = (2x)^2 - (9y)^2 = (2x - 9y)(2x + 9y) $
---
Part B: Expanding (Problems 17–26)
Use identity: $ (a + b)(a - b) = a^2 - b^2 $
17. $ (2x + 1)(2x - 1) = (2x)^2 - (1)^2 = 4x^2 - 1 $
18. $ (3x + 4)(3x - 4) = (3x)^2 - 4^2 = 9x^2 - 16 $
19. $ (4x + 5)(4x - 5) = (4x)^2 - 5^2 = 16x^2 - 25 $
20. $ (5x + 7)(5x - 7) = 25x^2 - 49 $
21. $ (6x + 8)(6x - 8) = 36x^2 - 64 $
22. $ (7x + 9)(7x - 9) = 49x^2 - 81 $
23. $ (8x + 10)(8x - 10) = 64x^2 - 100 $
24. $ (9x + 11)(9x - 11) = 81x^2 - 121 $
25. $ (10x + 12)(10x - 12) = 100x^2 - 144 $
26. $ (11x + 13)(11x - 13) = 121x^2 - 169 $
---
✔ Final Answer Summary
#### Factoring (1–16):
1. $ (3x - 2y)(3x + 2y) $
2. $ (2x - 9)(2x + 9) $
3. $ (6x - 11)(6x + 11) $ *(assuming typo)*
4. $ (7 - p)(7 + p) $
5. $ 25(2 - x)(2 + x) $
6. $ (y - 4x)(y + 4x) $
7. $ (5m - p)(5m + p) $
8. $ (4x - 3)(4x + 3) $
9. $ (6x - 5)(6x + 5) $
10. $ (5x - 7)(5x + 7) $
11. $ (2x - 3y)(2x + 3y) $
12. $ (5x - 4y)(5x + 4y) $
13. $ (4x - 5y)(4x + 5y) $
14. $ (7x - 4y)(7x + 4y) $
15. $ (11x - 10y)(11x + 10y) $
16. $ (2x - 9y)(2x + 9y) $
#### Expanding (17–26):
17. $ 4x^2 - 1 $
18. $ 9x^2 - 16 $
19. $ 16x^2 - 25 $
20. $ 25x^2 - 49 $
21. $ 36x^2 - 64 $
22. $ 49x^2 - 81 $
23. $ 64x^2 - 100 $
24. $ 81x^2 - 121 $
25. $ 100x^2 - 144 $
26. $ 121x^2 - 169 $
---
📌 Explanation:
All problems involve special products, specifically the difference of squares pattern:
$$
a^2 - b^2 = (a - b)(a + b)
$$
This is used both for factoring binomials and expanding products of conjugates.
The key is recognizing when an expression fits the form $ a^2 - b^2 $, and applying the formula accordingly.
Let me know if you'd like this formatted in a table or want to see any specific step in detail!
Parent Tip: Review the logic above to help your child master the concept of factoring special products worksheet.
