Answer:
The polynomial function is [tex]P(x) = x^4-35x^2+180x-416.[/tex]
Step-by-step explanation:
A polynomial function is completely determined by its roots, up to a constant factor. So, if we want that [tex]x_1=4[/tex], [tex]x_2=-8[/tex] and [tex]x_3=2+3i[/tex] be roots of the polynomial P, we can write it as
[tex]P(x)= (x-x_1)(x-x_2)(x-x_3)=(x-4)(x+8)(x-(2+3i)) = (x^2+4x-32)(x-(2+3i)).[/tex]
Now, notice that the factor [tex]x^2+4x-32[/tex] 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 [tex]x_3[/tex]: 2-3i. Then,
[tex]P(x) = (x^2+4x-32)(x-(2+3i))(x-(2-3i)).[/tex]
Expanding the above expression we obtain the desired polynomial
[tex]P(x) = x^4-35x^2+180x-416.[/tex]
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.
