I have been studying the Fundamental theorem of Algebra over the past few days, and I'm having a hard time finding the answers for my following question. I hope that someone will shine a light so I can move forward with my study.
From Wikipedia:
"The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero.
The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots. The equivalence of the two statements can be proven through the use of successive polynomial division."
My question is:
Is there such a thing as a polynomial with complex (imaginary part included) coefficients? Can a polynomial be created such as $f(x) = (x-2)(x-i)(x+2i) = x^3 + (2+i)x^2 + (2+2i)x + 4$? If not, is the reason for it the fact that the coefficients of the polynomial contain an imaginary part, or because it simply defies the complex conjugate root theorem?
Thank you.