What is the factored form of a^2 - 81?

What is the factored form of a^2 - 81?

• A. a^2 - 81
• B. (a + 9)^2
• C. (a + 9)(a - 9)
• D. (a - 9)^2

Answer: (C)

This is a difference of squares. When you have a^2 minus a perfect square, it factors as (a − 9)(a + 9) because 81 is 9^2. Expanding (a − 9)(a + 9) gives a^2 − 9a + 9a − 81, and the middle terms cancel, leaving a^2 − 81. The forms (a + 9)^2 and (a − 9)^2, when expanded, produce 18a or −18a as the middle term, plus 81, which is not the same as a^2 − 81. So the correct factorization is (a + 9)(a − 9).

This is a difference of squares. When you have a^2 minus a perfect square, it factors as (a − 9)(a + 9) because 81 is 9^2. Expanding (a − 9)(a + 9) gives a^2 − 9a + 9a − 81, and the middle terms cancel, leaving a^2 − 81. The forms (a + 9)^2 and (a − 9)^2, when expanded, produce 18a or −18a as the middle term, plus 81, which is not the same as a^2 − 81. So the correct factorization is (a + 9)(a − 9).