Final answer:
The polynomial 49x² - 81 is a difference of squares and can be factored as (7x + 9)(7x - 9), which is option A. The factoring process follows the identity a² - b² = (a + b)(a - b) where a and b are the square roots of the original terms. The correct answer is option a.
Step-by-step explanation:
The question is asking us to factor the polynomial 49x² - 81 completely. The given polynomial is a difference of squares because 49 (or 7²) and 81 (or 9²) are both perfect squares. We can apply the identity a² - b² = (a + b)(a - b), where a is the square root of the first term and b is the square root of the second term.
In this case, a = 7x and b = 9, so our factored form becomes (7x + 9)(7x - 9), which is option A. We don't have to consider the other options because this is the only option that follows the difference of squares pattern correctly. We must ensure that when factoring, each term is accounted for and that our result, when expanded, would return us to the original expression, 49x² - 81.
Step-by-step:
- Determine if the polynomial is a difference of squares: Both terms are perfect squares.
- Find the square root of each term: √49x² = 7x and √81 = 9.
- Apply the difference of squares formula: (a + b)(a - b).
- Write out the factored form using the square roots found in step 2: (7x + 9)(7x - 9).
- Confirm by expanding the factored form if needed to see if it matches the original polynomial.
The correct option is A. (7x + 9)(7x - 9).