The Fundamental Theorem of Algebra: If P(x) is a polynomial of degree n ≥ 1, then P(x) = 0 has exactly n roots, including multiple and complex roots.

In plain English, this theorem says that the degree of a polynomial equation tells you how many roots the equation will have.

A linear equation (degree 1) will have one root.
A quadratic equation (degree 2) will have two roots.
A cubic equation (degree 3) will have three roots.
An n^th degree polynomial equation will have n roots.

Read carefully to fully understand what this theorem is saying.
• It mentions "multiple" roots . . . so, if a polynomial has a repeated root, each repetition of the root is counted.
• It mentions "complex" roots . . . so, if a polynomial has a complex root, then the conjugate is also counted since complex roots come in conjugate pairs.
This theorem deals with complex roots, both real and non-real. Remember that all Real Numbers are a subset of the Complex Numbers.

Dealing with Quadratics:
Let's verify that the Fundamental Theorem of Algebra holds for quadratic polynomials.

The quadratic may have 2 distinct real roots. This graph crosses the x-axis in two locations. These graphs may open upward or downward.

It may appear that the quadratic has only one real root. But, it actually has one repeated root. This graph is tangent to the x-axis in one location (touching once).

The quadratic may have two non-real complex roots called a conjugate pair. This graph will not cross or touch the x-axis, but it has two complex (conjugate) roots.