Question

Question #5eda4

Answer 1

`f(x) = x^3 - 3x^2 + x - 3`

Explanation:

The complex conjugate root theorem states that if `a+bi` is a root of a polynomial, then so is `a-bi`. So, since `i` is a root, then `-i` will also be a root.

Since the polynomial has the zeros `3, i,` and `-i`, the factors of the polynomial will be `x-3`, `x-i`, and `x-(-i) = x + i`.

`f(x) = (x-3)(x-i)(x+i)`

Since `(a+b)(a-b) = a^2 - b^2`, we can rewrite the function as

`f(x) = (x-3)(x^2 - i^2)`

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

We can expand this further.

`f(x) = x^3 + x - 3x^2 - 3`

In standard form, the function is `f(x) = x^3 - 3x^2 + x - 3`.