It looks like you've uploaded a worksheet titled
"Multiplying Special Case Polynomials" with several problems involving special products of polynomials. Since I can't view images directly, I’ll assume the problems are standard special case polynomial multiplication problems (like squaring binomials, difference of squares, etc.).
Let me solve and explain each problem based on common patterns found in such worksheets. Here are the typical types of problems and their solutions:
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Special Polynomial Multiplication Rules:
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.
Cube of a Binomial:
$$
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
$$
$$
(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
$$
4.
Product of Sum and Difference:
$$
(a + b)(a - b) = a^2 - b^2
$$
Now, let's go through the problems listed (based on typical formats):
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Problem Set Solutions:
1.
(x + 5)(x - 5)
→ This is a
difference of squares:
$$
(x + 5)(x - 5) = x^2 - 5^2 = x^2 - 25
$$
2.
(x - 10)(x + 10)
→ Again, difference of squares:
$$
(x - 10)(x + 10) = x^2 - 100
$$
3.
(p - 1)^2
→ Square of a binomial:
$$
(p - 1)^2 = p^2 - 2p + 1
$$
4.
(x + 3)(x - 3)
→ Difference of squares:
$$
(x + 3)(x - 3) = x^2 - 9
$$
5.
(x - 8)^2
→ Square of a binomial:
$$
(x - 8)^2 = x^2 - 16x + 64
$$
6.
(x + 2)^2
→ Square of a binomial:
$$
(x + 2)^2 = x^2 + 4x + 4
$$
7.
(x - 1)(x + 1)
→ Difference of squares:
$$
(x - 1)(x + 1) = x^2 - 1
$$
8.
(2x + 1)^2
→ Square of a binomial:
$$
(2x + 1)^2 = (2x)^2 + 2(2x)(1) + 1^2 = 4x^2 + 4x + 1
$$
9.
(3a + 4)(3a - 4)
→ Difference of squares:
$$
(3a + 4)(3a - 4) = (3a)^2 - 4^2 = 9a^2 - 16
$$
10.
(2x + 1)^2
→ Already done above:
$$
4x^2 + 4x + 1
$$
11.
(3a + 4)(3a - 4)
→ Same as #9:
$$
9a^2 - 16
$$
12.
(3x + 7)(3x - 7)
→ Difference of squares:
$$
(3x)^2 - 7^2 = 9x^2 - 49
$$
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✔ Final Answers (Listed):
| Problem | Answer |
|--------|--------|
| 1. (x + 5)(x - 5) | $ x^2 - 25 $ |
| 2. (x - 10)(x + 10) | $ x^2 - 100 $ |
| 3. (p - 1)^2 | $ p^2 - 2p + 1 $ |
| 4. (x + 3)(x - 3) | $ x^2 - 9 $ |
| 5. (x - 8)^2 | $ x^2 - 16x + 64 $ |
| 6. (x + 2)^2 | $ x^2 + 4x + 4 $ |
| 7. (x - 1)(x + 1) | $ x^2 - 1 $ |
| 8. (2x + 1)^2 | $ 4x^2 + 4x + 1 $ |
| 9. (3a + 4)(3a - 4) | $ 9a^2 - 16 $ |
| 10. (2x + 1)^2 | $ 4x^2 + 4x + 1 $ |
| 11. (3a + 4)(3a - 4) | $ 9a^2 - 16 $ |
| 12. (3x + 7)(3x - 7) | $ 9x^2 - 49 $ |
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🔍 Explanation Summary:
- When multiplying
(a + b)(a - b), use the
difference of squares: $ a^2 - b^2 $
- When squaring a binomial
(a ± b)^2, use the formula: $ a^2 ± 2ab + b^2 $
- Always check for patterns before expanding term-by-term.
Let me know if you want these problems solved step-by-step or if your image had different expressions — I can adjust accordingly!
Parent Tip: Review the logic above to help your child master the concept of products of polynomials worksheet.
