Answer:
p(x) = 2(x³ + x² + 4x + 4)
Step-by-step explanation:
An imaginary root such as 2i must be joined by its complex conjugate, which here is -2i.
Thus, the zeros are -1, 2i and -2i.
The polynomial is p(x) = a(x + 1)(x - 2i)(x + 2i). This is of degree 3.
In expanded form, we have p(x) = a(x + 1)(x² + 4), or
= a(x³ + 4x + x² + 4), or, in standard form,
= a(x³ + x² + 4x + 4).
Since this must equal 8 when x = 0, 8 = a(4). Thus, a = 2.
The polynomial is thus p(x) = 2(x³ + x² + 4x + 4)
