Question #5eda4

1 Answer
Sep 4, 2017

Answer:

#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#.