The Fundamental Theorem of Algebra states that every polynomial of degree one or greater has at least one root in the complex number system (keep in mind that a complex number can be real if the imaginary part of the complex root is zero). A further theorem, in some cases referred to as the Linear Factorization Theorem, states that a polynomial of degree n has exactly n linear factors, and each can be written in the form (x - c), where c is a root. These n complex roots (possibly including some real roots) are counted with multiplicity. This much was explained in Roots of Higher Degree Polynomials.

In Roots of Higher Degree Polynomials, we discussed how a polynomial can be resolved into linear factors irreducible over the reals. The factors that are first-degree polynomials are real roots of the original polynomial. The factors that are second-degree polynomials can't be reduced using real numbers. With complex numbers, however, we can solve those quadratic equations which are irreducible over the reals, and we can then find each of the n roots of a polynomial of degree n.

A given quadratic equation ax2 + bx + c = 0 in which b2 -4ac < 0 has two complex roots: x = ,. Therefore, whenever a complex number is a root of a polynomial with real coefficients, its complex conjugate is also a root of that polynomial.

As an example, we'll find the roots of the polynomial x5 - x4 + x3 - x2 - 12x + 12.


x5 - x4 + x3 - x2 - 12x + 12 = 0    

(x - 1)(x4 + x2 - 12) = 0    

(x - 1)(x2 -3)(x2 + 4) = 0    

(x - 1)(x + )(x - )(x2 + 4) = 0    

(x - 1)(x + )(x - )(x + 2ı)(x - 2ı) = 0    

The fifth-degree polynomial does indeed have five roots; three real, and two complex.