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Answer:
The polynomial function is
Step-by-step explanation:
A polynomial function is completely determined by its roots, up to a constant factor. So, if we want that ,
and
be roots of the polynomial P, we can write it as
Now, notice that the factor has real coefficients, while the other one don't. So, we need to ‘‘eliminate’’ the complex coefficients that will appear. This can be done adding other complex root to the polynomial: the conjugate of
: 2-3i. Then,
Expanding the above expression we obtain the desired polynomial
Recall that P(x) must have three roots, and this implies that P has at least degree 3. As we had to add a new root in order to obtain real coefficients, the degree of P must be at least 4. With this reasoning we assure the minimal degree.