Question

I just need to know the answer and some explanation please

Answer 1

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:

`(x+5)(x+3)(x-1)(x-3)(x-4)`

Next, it must have a factor that produces a non-real zero:

`\begin{gathered} x^2+2=0 \\ \implies x=\pm2i \end{gathered}`

One example of a factor that produces a non-real zero is x²+2.

Adding this factor to the previous ones, the polynomial becomes:

`f(x)=(x+5)(x+3)(x-1)(x-3)(x-4)(x^2+2)`

The polynomial is of a degree of 7 as required.

The sketch of the polynomial is given below: