Question

Facto. If the polynomial is not factorable, prime.

𝑥^5 −16⁢𝑥

Answer 1

Factor out x: x^5 − 16x = x(x^4 − 16).

Recognize a difference of squares: x^4 − 16 = (x^2 − 4)(x^2 + 4).

Further factor x^2 − 4 = (x − 2)(x + 2).

So over the integers:
x^5 − 16x = x(x − 2)(x + 2)(x^2 + 4).

(x^2 + 4 is irreducible over the integers/rationals; over the complex numbers it factors as (x − 2i)(x + 2i).)