Answer:
f(x)=(x+5)(x+3)(x-1)(x-3)(x-4)(x²+2)
Explanation:
Given that the polynomial must have zeros at -5, -3, 1, 3, and 4, some of the factors are:
[tex](x+5)(x+3)(x-1)(x-3)(x-4)[/tex]
Next, it must have a factor that produces a non-real zero:
[tex]\begin{gathered} x^2+2=0 \\ \implies x=\pm2i \end{gathered}[/tex]
One example of a factor that produces a non-real zero is x²+2.
Adding this factor to the previous ones, the polynomial becomes:
[tex]f(x)=(x+5)(x+3)(x-1)(x-3)(x-4)(x^2+2)[/tex]
The polynomial is of a degree of 7 as required.
The sketch of the polynomial is given below:
