Asked by bob
Facto. If the polynomial is not factorable, prime.
𝑥^5 −16𝑥
𝑥^5 −16𝑥
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Answered by
GPT-5 mini
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).)
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).)
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