I'm in college precalculus and I am curious. I am given a polynomial function $f(x) = x^4 -7x^3 + 18x^2 -22x + 12$ and given that $(1-i)$ is a zero.
I understand that to find the real zeros (factors, solutions, roots) you synthetically divide the polynomial by $(1-i)$, and then divide that quotient by $(1+i)$ which yields a polynomial factorable to obtain real roots. Then I multiplied $(x-(1-i))$ by $(x-(1+i))$ to obtain another polynomial with real roots $(x^2-2x+2)$.
All in all, the original function in factored form is $f(x)=(x^2-2x+2)(x-2)(x-3)$ which seems to match the graph of the given function.
So, my question is... what is the relevance of the problem's given root $(1-i)$? If it doesn't appear on the x,y coordinate plane as a zero, what the heck does it mean? Why is it important and how would you even find this complex root, if say all you were given was $f(x) = x^4 -7x^3 + 18x^2 -22x + 12$ (without being told in the setup that $(1-i)$ was a root)?
I know I can find the fully factored form of the original function with methods not involving this given complex root, so I mean, what's the point? Again, I'm only in precalculus, but I'm assuming they are having us work on these things for a reason (but that reason is not provided in my text).