Fundamental Theory of Algebra

The fundamental theory of algebra states that any polynomial of degree n has n roots. For example, the function  has seven roots, because the highest exponent is seven. Those roots are . Five of these roots are real, but two are unreal. This can be seen in the graph:

Again, this function has five, real, visible roots and two unreal ones that cannot be easily seen on the graph. The roots we cannot see as x-intercepts can be assumed to be imaginary. This can be demonstrated in the translation of the function .

Four real roots, zero imaginary roots. 

Two reals roots, two imaginary roots.

Zero real roots, four imaginary roots.

As you probably have noticed, the roots in this function come in pairs. Real roots only come in pairs in a function that is symmetrical about the y-axis, but imaginary roots always come in pairs, no matter the shape. This is because imaginary roots come in conjugate pairs. For example, if F(2+3i)=0 then, F(2-3i)=0.