Question

Completely factor P(x)=x^4+x^3+4x^2+16x-192 into linear factors, given that -4i is a zero.

P(x)=?

Answer 1

he complete factorization of P(x) is (x^2 + 16)(x - 3)(x + 4).

Step-by-step explanation:

To factor the polynomial P(x) = x^4 + x^3 + 4x^2 + 16x - 192, we know that -4i is a zero.

Since -4i is a zero, its conjugate 4i must also be a zero.

We can now use synthetic division to find the remaining quadratic factor.

The synthetic division gives us (x^2 + 16).

Now, we can rewrite P(x) as (x^2 + 16)(x^2 + x - 12).

Further factoring, we get (x^2 + 16)(x - 3)(x + 4).

Therefore, the complete factorization of P(x) is (x^2 + 16)(x - 3)(x + 4).