2.5 - Fundamental Theorem of Algebra and Complex Conjugate Pairs

Section 2.5 addresses the fundamental theorem of algebra as well as complex conjugate pairs.

Fundamental Theorem of Algebra

As quoted from the book, the theorem states...

"If f(x) is a polynomial of degree n, where n > 0, f has at least one zero in the complex number system."

The theorem also states that given an nth degree polynomial, the polynomial has n amount of roots, or zeros.

Example

Given the equation:

The polynomial is to the 1st degree, so we can conclude that there is one root, or zero. We can solve and confirm.

We solve and figure out that the equation has exactly one root: x = 2.

Yet we must remember that not all zeros are x-intercepts. Some zeros are found out to be imaginary.

Example 2

Given the equation:

We see that the polynomial is of the fourth degree. We can then conclude that it has four zeros. Let's solve and confirm.

Which is then split into two parts:

We can see that there are four zeros. Yet not all of them can be graphed. Two of them are imaginary. But, remember, they are still zeros!

Complex Conjugate Pairs

As seen in the previous section, roots of functions can be real, or imaginary. And as we also saw, the number of roots depends on the degree of the function. We will now learn that when an equation contains an imaginary root of form:

It will also contain the root:

From this we can conclude that there can only be an even number of imaginary roots. 

Also, be sure to remember that when eliminating imaginary numbers from denominators, multiply them by their conjugate, using the above form.