Final answer:
The polynomial x^4 - 36 is factored by recognizing it as a difference of squares, resulting in the factored form (x^2 + 6)(x^2 - 6).
Step-by-step explanation:
The student has asked us to factor the polynomial x^4 - 36. This can be seen as a difference of squares because 36 is a perfect square (6^2). The difference of squares formula states that a^2 - b^2 = (a + b)(a - b). Applying this to the given polynomial, we can rewrite x^4 as (x^2)^2 and 36 as (6)^2. Therefore, the factored form of the polynomial x^4 - 36 is (x^2 + 6)(x^2 - 6). Furthermore, the expression x^2 - 6 can also be factored as a difference of squares since 6 can be written as (√6)^2.
However, x^2 - 6 does not represent a difference of perfect squares in terms of integers or rationals, but it could be factored over the real numbers as (x + √6)(x - √6). If we were looking only for rational factorizations, we'd stop at (x^2 + 6)(x^2 - 6).