Answer :
To factor the expression [tex]\(9z^2 - 169\)[/tex], we can recognize it as a difference of squares. A difference of squares is a special pattern that looks like [tex]\(a^2 - b^2\)[/tex], and it can be factored into [tex]\((a - b)(a + b)\)[/tex].
Here's how we can apply this pattern:
1. Identify the squared terms:
- Notice that [tex]\(9z^2\)[/tex] is a perfect square because it can be rewritten as [tex]\((3z)^2\)[/tex].
- Notice that [tex]\(169\)[/tex] is also a perfect square because it can be rewritten as [tex]\(13^2\)[/tex].
2. Write the expression as a difference of squares:
- Since [tex]\(9z^2 = (3z)^2\)[/tex] and [tex]\(169 = 13^2\)[/tex], we can write the expression as [tex]\((3z)^2 - 13^2\)[/tex].
3. Apply the difference of squares formula:
- The formula for a difference of squares is [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].
- Here, [tex]\(a = 3z\)[/tex] and [tex]\(b = 13\)[/tex].
4. Factor the expression:
- Substitute the values into the formula: [tex]\((3z - 13)(3z + 13)\)[/tex].
Therefore, the factored form of the expression [tex]\(9z^2 - 169\)[/tex] is [tex]\((3z - 13)(3z + 13)\)[/tex].
Here's how we can apply this pattern:
1. Identify the squared terms:
- Notice that [tex]\(9z^2\)[/tex] is a perfect square because it can be rewritten as [tex]\((3z)^2\)[/tex].
- Notice that [tex]\(169\)[/tex] is also a perfect square because it can be rewritten as [tex]\(13^2\)[/tex].
2. Write the expression as a difference of squares:
- Since [tex]\(9z^2 = (3z)^2\)[/tex] and [tex]\(169 = 13^2\)[/tex], we can write the expression as [tex]\((3z)^2 - 13^2\)[/tex].
3. Apply the difference of squares formula:
- The formula for a difference of squares is [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].
- Here, [tex]\(a = 3z\)[/tex] and [tex]\(b = 13\)[/tex].
4. Factor the expression:
- Substitute the values into the formula: [tex]\((3z - 13)(3z + 13)\)[/tex].
Therefore, the factored form of the expression [tex]\(9z^2 - 169\)[/tex] is [tex]\((3z - 13)(3z + 13)\)[/tex].
