Final answer:
To find the polynomial of least degree with rational coefficients and given roots -1 and 4i, one must also include the conjugate -4i as a root. The resulting polynomial is p(x) = x^3 + x^2 + 16x + 16.
Step-by-step explanation:
To write the polynomial function of the least degree with rational coefficients that has the given roots of -1 and 4i, we should first recognize that if a polynomial has rational coefficients and a complex root, then its conjugate must also be a root. This means we need to include the conjugate of 4i, which is -4i, as a root.
So, the roots of the polynomial are -1, 4i, and -4i. The polynomial is formed by multiplying the factors associated with these roots:
(x - (-1))(x - 4i)(x + 4i)
This simplifies to:
(x + 1)(x^2 + 16)
Finally, we multiply out these factors to get the polynomial:
p(x) = x^3 + x^2 + 16x + 16
This is the polynomial function of least degree with rational coefficients that has the given roots.