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Let's go through each problem and factor the polynomials. These are all based on factoring differences or sums of squares, which are common patterns in algebra.

### 1. \( x^2 - 36 \)
This is a difference of squares:
\[
x^2 - 36 = (x - 6)(x + 6)
\]

### 2. \( a^2b^2 - 100 \)
This is also a difference of squares:
\[
a^2b^2 - 100 = (ab - 10)(ab + 10)
\]

### 3. \( 16y^2 - 25 \)
Another difference of squares:
\[
16y^2 - 25 = (4y - 5)(4y + 5)
\]

### 4. \( y^2 + 49 \)
This is a sum of squares, which typically can't be factored over the real numbers. The expression is prime.

### 5. \( 9a^2 - 121 \)
This is a difference of squares:
\[
9a^2 - 121 = (3a - 11)(3a + 11)
\]

### 6. \( 196 - x^2 \)
This is another difference of squares:
\[
196 - x^2 = (14 - x)(14 + x)
\]

### 7. \( 4y^2 - 81 \)
This is a difference of squares:
\[
4y^2 - 81 = (2y - 9)(2y + 9)
\]

### 8. \( 36c^2 - d^2 \)
This is a difference of squares:
\[
36c^2 - d^2 = (6c - d)(6c + d)
\]

### 9. \( 16m^2 - p^2 \)
This is a difference of squares:
\[
16m^2 - p^2 = (4m - p)(4m + p)
\]

### 10. \( m^2 - 64n^2 \)
Another difference of squares:
\[
m^2 - 64n^2 = (m - 8n)(m + 8n)
\]

### 11. \( 1 - 4a^2 \)
This is a difference of squares:
\[
1 - 4a^2 = (1 - 2a)(1 + 2a)
\]

### 12. \( 9v^4 - 100 \)
This is a difference of squares, with the \( v^4 \) term being a square as well:
\[
9v^4 - 100 = (3v^2 - 10)(3v^2 + 10)
\]

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Would you like further details on any of these factorizations, or do you have any questions?

Here are some follow-up questions that could deepen your understanding:
1. What is the general formula for factoring a difference of squares?
2. Why can't the sum of squares typically be factored over the real numbers?
3. Can you apply FOIL to check your answers for each factorization?
4. How would these problems change if the coefficients in front of the squares were not perfect squares themselves?
5. What happens when you try factoring trinomials compared to these binomials?

**Tip:** Always check your factoring by multiplying the factors back together to verify the original polynomial!

Factor each polynomial, if possible. Check your answer using FOIL.

Learn how to factor polynomials such as x^2 - 36, a^2b^2 - 100, and more. This problem set includes common algebraic techniques such as factoring differences of squares and checking solutions using the FOIL method. Suitable for Grades 8-10.

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