Question

Which expression represents 625p4−16
when factored completely over the complex numbers?

(5p−2i)2(5p+2i)2

(25p2+4i)(25p2−4i)
(25p2+4i)(5p−2i)(5p+2i)
(5p−2)(5p+2)(5p−2i)(5p+2i)

Answer 1

Answer: the fourth option (5p - 2)(5p + 2) (5p - 2i) (5p + 2i)

Explanation:

1) Express the given expression as the difference of two squares:

625p⁴ - 16 = (25p²)² - 4²

2) Factor as the product of a sum times its difference:

(25p²)² - 4² = (25p² - 4) (25p² + 4)

3) Factor each binomial (again square of a difference and sum times its difference):

25p² - 4 = (5p)² - 2² = (5p - 2)(5p + 2)

25p² + 4 = (5p)² - (2i)² = (5p - 2i)(5p + 2i)

4) Therefore, the 4 factors are:

(5p - 2)(5p + 2) (5p - 2i) (5p + 2i)