6-6 The Fundamental Theorem of Algebra
Basically, the Fundamental Theorem of Algebra says that the number of complex
roots (TOTAL # of roots including real and complex number roots) is equal to the
highest degree of the polynomial (e.g. a 4th degree polynomial has 4 “complex” roots
altogether, a 3rd degree polynomial has 3 complex roots, etc.
Now, combining this theorem with the imaginary number theorem, since imaginary
numbers come in pairs (a + bi and its conjugate, a – bi), then if you have a 4th degree
polynomial, you either have all 4 roots as real roots, or 2 real roots or 0 real roots.
You cannot have an odd number of real roots (1 or 3) in this case, because then you
would have 3 or 1 imaginary roots which is impossible since imaginary roots come
in pairs.