what polynomial identities would result from dividing x^2-a^2 by x-a. Enter the option of the coorect answer

Question

option 1: x^2-a^2=(x-a)(x+a)
option 2: x^2-a^2=(x-a)(x-a)
option 3: x^2-a^2(x+a)(x+a)

Answer 1

To find the polynomial identity that results from dividing \( x^2 - a^2 \) by \( x - a \), we can first recall the difference of squares identity:

\[ x^2 - a^2 = (x - a)(x + a) \]

Now, let's evaluate the options:

• Option 1: \( x^2 - a^2 = (x - a)(x + a) \) - This matches the difference of squares identity and is correct.

• Option 2: \( x^2 - a^2 = (x - a)(x - a) \) - This is incorrect as it does not correspond to the identity for the difference of squares.

• Option 3: \( x^2 - a^2 = (x + a)(x + a) \) - This is also incorrect since it doesn't correspond to the factorization of \( x^2 - a^2 \).

Therefore, the correct answer is:

Option 1: \( x^2 - a^2 = (x - a)(x + a) \).