To factor the given expression [tex]\( 81 y^2 - 100 \)[/tex], follow these steps:
1. Express in Standard Form:
The given expression is [tex]\( 81 y^2 - 100 \)[/tex].
2. Identify as a Difference of Squares:
Notice that [tex]\( 81 y^2 - 100 \)[/tex] is a difference of squares. Recall the formula for factoring a difference of squares:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
3. Rewrite Each Term as a Square:
Recognize that [tex]\( 81 y^2 = (9y)^2 \)[/tex] and [tex]\( 100 = 10^2 \)[/tex]. Thus, the expression can be rewritten as:
[tex]\[
(9 y)^2 - 10^2
\][/tex]
4. Apply the Difference of Squares Formula:
Using the difference of squares formula:
[tex]\[
(9 y)^2 - 10^2 = (9 y - 10)(9 y + 10)
\][/tex]
5. Confirm the Factors:
The expression [tex]\( 81 y^2 - 100 \)[/tex] factors to [tex]\((9 y - 10)(9 y + 10)\)[/tex].
Therefore, the correct factorization is obtained, and the correct answer among the given choices is:
[tex]\[
(9 y + 10)(9 y - 10)
\][/tex]
