Question

Using the given zero, find one other zero of f(x). Explain the process you used to find your solution.

1 - 6i is a zero of f(x) = x4 - 2x3 + 38x2 - 2x + 37.

Answer 1

given
\[x = 1-6i\]
then
\[x-1 = -6i\]
\[(x-1)^{2} = (-6i)^{2}\]
\[x^{2} -2x+1 = -36\]
\[x^{2} -2x +37 = 0\]
Now this is one of the factors of polynomial...to get other factor divide f(x) by this factor

x^2 + 1
____________________________
x^2 -2x +37 | x^4 -2x^3 +38x^2 -2x+37
-(x^4 -2x^3+37x^2)
------------------
x^2 -2x +37
-(x^2 -2x +37)
--------------
0

to get other zeros, set factor equal to 0
\[x^{2} +1 = 0\]
\[x^{2} = -1\]
\[x = \pm i\]