Final answer:
The fourth factor of the polynomial is -2i because non-real roots must occur in conjugate pairs. The polynomial is thus f(x) = (x + 1)(x - 3)(x2 + 4).
Step-by-step explanation:
If a fourth degree polynomial has the factors -1, 3, and 2i, then by the Complex Conjugate Root Theorem, the fourth factor must be the complex conjugate of 2i, which is -2i since non-real roots of polynomials with real coefficients occur in conjugate pairs. So, the fourth factor of the polynomial is -2i. Therefore, a fourth degree polynomial with these roots could be written as f(x) = (x + 1)(x - 3)(x - 2i)(x + 2i), which simplifies to f(x) = (x + 1)(x - 3)(x2 + 4).
A fourth degree polynomial has the factors -1, 3, and 2i. To find the 4th factor, we can use a process called factoring by grouping. First, we'll set up the polynomial using the given factors:
(x + 1)(x - 3)(x - 2i)(x + 2i)
Now, we can use the complex conjugate property to simplify the expression. The complex conjugate of 2i is -2i, so we can rewrite the expression as:
(x + 1)(x - 3)(x - 2i)(x + 2i) = (x + 1)(x - 3)(x^2 + 4)
Therefore, the 4th factor is x^2 + 4.