What is the complete factorization of the polynomial function over the set of complex numbers? f(x)=x3−4x2+4x−16 enter your answer in the box.

To factor the expression, we perform the following steps:

Separate the expression:

We take common factor:

We take common factor (x-4):

We zero each expression and solve:

 For the first expression:

x-4 = 0

x = 4 real root

For the second expression:

complex roots

and

and

x = 2i and x = -2i

We already have the real root and the complex roots, now we factor:

if the eqn of the parabola is f(x) = (x-4)(x+2), we need to mult. this out to find the coordinates of the vertex.

then f(x) = x^2 + 2x - 4x - 8, and f(x) = x^2 - 2x - 8.

one way in which you can find the vertex is to calculate x = -b / (2a).   here, that comes to

        )

x = = 1.   f(1) = 1^2 - 2(1) - 8 = - 9.   vertex is at (1, -9).

        2(1) 

find another point on the curve:   let x=0, for which y=-8.   then (0, -8) is on the curve.   plot the vertex and this point and draw a smooth parabola thru both.

hope this