Final answer:
To completely factor the polynomial p(x)=x⁴+2x³+8x²+32x-128, we can start by using the given zero -4i and finding its conjugate 4i. Then, we can divide the polynomial by these factors and the remaining linear factors to get the completely factored form.
Step-by-step explanation:
To factor the polynomial p(x) = x⁴ + 2x³ + 8x² + 32x - 128, we can start by using the given zero -4i. Since -4i is a zero, we know that its conjugate 4i is also a zero. So, the factors for the polynomial are (x + 4i), (x - 4i), and two other linear factors.
We can use long division or synthetic division to divide the polynomial by (x + 4i)(x - 4i) to find the remaining linear factors. After dividing, we get p(x) = (x + 4i)(x - 4i)(x - 4)(x + 8).
Therefore, the completely factored form of p(x) is p(x) = (x + 4i)(x - 4i)(x - 4)(x + 8).