Question

What is the complete factorization of the polynomial function over the set of complex numbers? f(x)=x3+3x2+16x+48

Answer 1

The polynomial function f(x) = x^3 + 3x^2 + 16x + 48 can be factored completely over the set of complex numbers as (x - 2)(x^2 + 5x + 24).

Step-by-step explanation:

The polynomial function f(x) = x^3 + 3x^2 + 16x + 48 can be factored completely over the set of complex numbers using methods such as synthetic division or factoring by grouping.

By applying synthetic division, we can find that (x - 2) is a factor of the polynomial. Using synthetic division again, we can factor the polynomial as f(x) = (x - 2)(x^2 + 5x + 24).

Now, we can factor the quadratic expression x^2 + 5x + 24 by factoring the trinomial or using the quadratic formula. However, in this case, the quadratic expression cannot be factored further over the set of complex numbers, so x^2 + 5x + 24 is the complete factorization of the polynomial function.

Answer 2

(x+3) (x-4i) (x+4i) is the answer, just took the quiz :)