College Algebra/Polynomial Functions

Question

Suppose a polynomial function of degree 4 with rational coefficients has zeros of -4, 1, and 2i. Find the remaining zero. Write the polynomial function as a product of linear factors.

Suppose a polynomial function of degree 5 with rational coefficients has 4, -2i, and 4+i as zeros. Find the remaining zeros.

Answer 1

First question: The remaining zero must be the complex conjugate of 2i, which is -2i.

f(x) = A(x + 4)(x - 1)(x - 2i)(x + 2i),

where A is real, rational, and A ≠ 0.

Second question: The remaining zeros are the complex conjugates of -2i and 4 + i, which are 2i and 4 - i, respectively.