To factor the polynomial [tex]\(121y^2 - 144\)[/tex], we can follow these steps:
1. Identify the form of the polynomial:
The given polynomial is a quadratic expression of the form [tex]\(ax^2 + bx + c\)[/tex]. Specifically, in this case, it looks like a difference of squares, which is a special form.
2. Recognize it as a difference of squares:
The difference of squares formula is [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex]. We need to express [tex]\(121y^2 - 144\)[/tex] in this form.
3. Rewrite each term as a square:
- [tex]\(121y^2\)[/tex] can be written as [tex]\((11y)^2\)[/tex], because [tex]\(11y \cdot 11y = 121y^2\)[/tex].
- [tex]\(144\)[/tex] can be written as [tex]\(12^2\)[/tex], because [tex]\(12 \cdot 12 = 144\)[/tex].
4. Apply the difference of squares formula:
Now we can express [tex]\(121y^2 - 144\)[/tex] as [tex]\((11y)^2 - 12^2\)[/tex].
Using the formula [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex], we set [tex]\(a = 11y\)[/tex] and [tex]\(b = 12\)[/tex].
Therefore, [tex]\((11y)^2 - 12^2 = (11y - 12)(11y + 12)\)[/tex].
5. Write the factored form:
The factored form of [tex]\(121y^2 - 144\)[/tex] is:
[tex]\[
(11y - 12)(11y + 12)
\][/tex]
So, the factored form of the polynomial [tex]\(121y^2 - 144\)[/tex] is [tex]\((11y - 12)(11y + 12)\)[/tex].
