directions: factor each polynomial, if possible. check your answer using foil.
1. ( x^2 - 36 )
2. ( a^2b^2 - 100 )
3. ( 16y^2 - 25 )
4. ( y^2 + 49 )
5. ( 9a^2 - 121 )
6. ( 196 - x^2 )
7. ( 4y^2 - 81 )
8. ( 36c^2 - d^2 )
9. ( 16m^2 - p^2 )
10. ( m^2 - 64n^2 )
11. ( 1 - 4a^2 )
12. ( 9v^4 - 100 )

Question

Accepted Answer

### Problem 1: $ x^2 - 36 $
# Explanation:
## Step 1: Identify the form
This is a difference of squares, $ a^2 - b^2 = (a - b)(a + b) $. Here, $ a = x $, $ b = 6 $ (since $ 6^2 = 36 $).
## Step 2: Apply the formula
$ x^2 - 36 = (x - 6)(x + 6) $
## Step 3: Check with FOIL
$ (x - 6)(x + 6) = x^2 + 6x - 6x - 36 = x^2 - 36 $, which matches the original polynomial.
# Answer: $ (x - 6)(x + 6) $

### Problem 2: $ a^2b^2 - 100 $
# Explanation:
## Step 1: Identify the form
This is a difference of squares, $ a^2 - b^2 = (a - b)(a + b) $. Here, $ a = ab $ (since $ (ab)^2 = a^2b^2 $), $ b = 10 $ (since $ 10^2 = 100 $).
## Step 2: Apply the formula
$ a^2b^2 - 100 = (ab - 10)(ab + 10) $
## Step 3: Check with FOIL
$ (ab - 10)(ab + 10) = a^2b^2 + 10ab - 10ab - 100 = a^2b^2 - 100 $, which matches the original polynomial.
# Answer: $ (ab - 10)(ab + 10) $

### Problem 3: $ 16y^2 - 25 $
# Explanation:
## Step 1: Identify the form
This is a difference of squares, $ a^2 - b^2 = (a - b)(a + b) $. Here, $ a = 4y $ (since $ (4y)^2 = 16y^2 $), $ b = 5 $ (since $ 5^2 = 25 $).
## Step 2: Apply the formula
$ 16y^2 - 25 = (4y - 5)(4y + 5) $
## Step 3: Check with FOIL
$ (4y - 5)(4y + 5) = 16y^2 + 20y - 20y - 25 = 16y^2 - 25 $, which matches the original polynomial.
# Answer: $ (4y - 5)(4y + 5) $

### Problem 4: $ y^2 + 49 $
# Explanation:
This is a sum of squares, $ y^2 + 7^2 $. In real numbers, a sum of squares (where the terms are not multiplied by a common factor) does not factor further. So it is prime (cannot be factored over the real numbers).
# Answer: Prime (cannot be factored)

### Problem 5: $ 9a^2 - 121 $
# Explanation:
## Step 1: Identify the form
This is a difference of squares, $ a^2 - b^2 = (a - b)(a + b) $. Here, $ a = 3a $ (since $ (3a)^2 = 9a^2 $), $ b = 11 $ (since $ 11^2 = 121 $).
## Step 2: Apply the formula
$ 9a^2 - 121 = (3a - 11)(3a + 11) $
## Step 3: Check with FOIL
$ (3a - 11)(3a + 11) = 9a^2 + 33a - 33a - 121 = 9a^2 - 121 $, which matches the original polynomial.
# Answer: $ (3a - 11)(3a + 11) $

### Problem 6: $ 196 - x^2 $
# Explanation:
## Step 1: Rewrite as difference of squares
$ 196 - x^2 = 14^2 - x^2 $, which is a difference of squares, $ a^2 - b^2 = (a - b)(a + b) $. Here, $ a = 14 $, $ b = x $.
## Step 2: Apply the formula
$ 14^2 - x^2 = (14 - x)(14 + x) $
## Step 3: Check with FOIL
$ (14 - x)(14 + x) = 196 + 14x - 14x - x^2 = 196 - x^2 $, which matches the original polynomial.
# Answer: $ (14 - x)(14 + x) $

### Problem 7: $ 4y^2 - 81 $
# Explanation:
## Step 1: Identify the form
This is a difference of squares, $ a^2 - b^2 = (a - b)(a + b) $. Here, $ a = 2y $ (since $ (2y)^2 = 4y^2 $), $ b = 9 $ (since $ 9^2 = 81 $).
## Step 2: Apply the formula
$ 4y^2 - 81 = (2y - 9)(2y + 9) $
## Step 3: Check with FOIL
$ (2y - 9)(2y + 9) = 4y^2 + 18y - 18y - 81 = 4y^2 - 81 $, which matches the original polynomial.
# Answer: $ (2y - 9)(2y + 9) $

### Problem 8: $ 36c^2 - d^2 $
# Explanation:
## Step 1: Identify the form
This is a difference of squares, $ a^2 - b^2 = (a - b)(a + b) $. Here, $ a = 6c $ (since $ (6c)^2 = 36c^2 $), $ b = d $.
## Step 2: Apply the formula
$ 36c^2 - d^2 = (6c - d)(6c + d) $
## Step 3: Check with FOIL
$ (6c - d)(6c + d) = 36c^2 + 6cd - 6cd - d^2 = 36c^2 - d^2 $, which matches the original polynomial.
# Answer: $ (6c - d)(6c + d) $

### Problem 9: $ 16m^2 - p^2 $
# Explanation:
## Step 1: Identify the form
This is a difference of squares, $ a^2 - b^2 = (a - b)(a + b) $. Here, $ a = 4m $ (since $ (4m)^2 = 16m^2 $), $ b = p $.
## Step 2: Apply the formula
$ 16m^2 - p^2 = (4m - p)(4m + p) $
## Step 3: Check with FOIL
$ (4m - p)(4m + p) = 16m^2 + 4mp - 4mp - p^2 = 16m^2 - p^2 $, which matches the original polynomial.
# Answer: $ (4m - p)(4m + p) $

### Problem 10: $ m^2 - 64n^2 $
# Explanation:
## Step 1: Identify the form
This is a difference of squares, $ a^2 - b^2 = (a - b)(a + b) $. Here, $ a = m $, $ b = 8n $ (since $ (8n)^2 = 64n^2 $).
## Step 2: Apply the formula
$ m^2 - 64n^2 = (m - 8n)(m + 8n) $
## Step 3: Check with FOIL
$ (m - 8n)(m + 8n) = m^2 + 8mn - 8mn - 64n^2 = m^2 - 64n^2 $, which matches the original polynomial.
# Answer: $ (m - 8n)(m + 8n) $

### Problem 11: $ 1 - 4a^2 $
# Explanation:
## Step 1: Rewrite as difference of squares
$ 1 - 4a^2 = 1^2 - (2a)^2 $, which is a difference of squares, $ a^2 - b^2 = (a - b)(a + b) $. Here, $ a = 1 $, $ b = 2a $.
## Step 2: Apply the formula
$ 1^2 - (2a)^2 = (1 - 2a)(1 + 2a) $
## Step 3: Check with FOIL
$ (1 - 2a)(1 + 2a) = 1 + 2a - 2a - 4a^2 = 1 - 4a^2 $, which matches the original polynomial.
# Answer: $ (1 - 2a)(1 + 2a) $

### Problem 12: $ 9v^4 - 100 $
# Explanation:
## Step 1: Identify the form
This is a difference of squares, $ a^2 - b^2 = (a - b)(a + b) $. Here, $ a = 3v^2 $ (since $ (3v^2)^2 = 9v^4 $), $ b = 10 $ (since $ 10^2 = 100 $).
## Step 2: Apply the formula
$ 9v^4 - 100 = (3v^2 - 10)(3v^2 + 10) $
## Step 3: Check with FOIL
$ (3v^2 - 10)(3v^2 + 10) = 9v^4 + 30v^2 - 30v^2 - 100 = 9v^4 - 100 $, which matches the original polynomial.
# Answer: $ (3v^2 - 10)(3v^2 + 10) $

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QUESTION IMAGE

Question

Explanation:

Problem 1: \( x^2 - 36 \)

Step 1: Identify the form

Step 2: Apply the formula

Step 3: Check with FOIL

Step 1: Identify the form

Step 2: Apply the formula

Step 3: Check with FOIL

Step 1: Identify the form

Step 2: Apply the formula

Step 3: Check with FOIL

Answer:

Problem 2: \( a^2b^2 - 100 \)