2.5 The Fundamental Theory of Algebra

In the complex number system, every nth-degree polynomial function has exactly n zeros. This is derived from the Fundamental Theorem of Algebra

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

The fundamental theorem of algebra also gives you the linear factorization theorem.

Linear Factorization Theorem - If f(x) is a polynomial of degree n where n > 0, f has precisely n linear factors


Where c1, c2, . . . , cn are complex numbers

Real and Complex Zeros of Polynomial Functions

Example:

 

This polynomial can be factored into:

 

This shows that the zeros are
(x-2) with a multiplicity of two, (x-2i) and (x+2i)

Notice that (x-2) is listed twice and (x-2i) and (x+2i) are both complex zeros.

Remember:
***The n zeros of a polynomial function can be real or complex and they may be repeated***
***Imaginary roots always come in conjugate pairs***

Finding the Zeros of a Polynomial Function

Example: 



The possible rational zeros of this polynomial function are ±1, ±2, ±4, and ±8 
(Find these by taking factors of the constant divided by the factors of the leading coefficient. The leading coefficient is 1 so we only need the factors of the constant)

By graphing this function, you can see that -2 and 1 are both real zeros. 
***Only the real zeros are shown on the graph as x-intercepts***
After finding the real zeros, you can find the imaginary zeros by using synthetic division.
Using -2 and 1 twice you are left with:



To find the zeros we need to factor this





This gives us the remaining non-real zeros. All five zeros of this polynomial are:

1, 1, -2, 2i, and -2i

Conjugate Pairs

In the last example two of the zeros were -2i and 2i. This is because all imaginary roots must come in conjugate pairs, in the form a + bi and a - bi

In a polynomial function with real coefficients if a + bi is a zero of the function and b is not 0 then the conjugate a - bi must also be a zero of the function.

Finding a Polynomial with Given Zeros

Given the degree of a polynomial function with real coefficients and its zeros we can easily determine the polynomial.

Example:

A third degree polynomial function with real coefficients has 2 and 4i as zeros.

Given 4i as a zero we know that -4i must also be a zero. Because the polynomial function is a third degree function we now have all the zeros: 2, 4i, and -4i.

We can write these out as:



To get the original polynomial we can multiply these factors.



This results in:



Factoring A Polynomial

Factors of a Polynomial - Every Polynomial of degree n > 0 with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros.

Example:



This can be factored into two quadratic polynomials



This can be further factored to include imaginary numbers



Zeros of a Polynomial Function

Given one complex zero of a polynomial function we can find the remaining zeros.

Example:

5i is a zero of   

Because 5i is a complex number, we know that -5i must also be a zero of this polynomial because all complex zeros come in conjugate pairs.

We can multiply these together to get part of the polynomial:






We can now use long division using this to find the remaining zero.

                                       
                                                                       
/
              
                                                          
                     
                     
                                                                   
                                               

This gives us 



The zeros of this polynomial are: