12. APPLICATIONS TO THE FUNDAMENTAL THEOREM  OF ALGEBRA AND OTHERS

12.1   Fundamental Theorem of Algebra

Theorem 1. (Fundamental Theorem of Algebra). A non-constant polynomial  has at least one root.

Proof. First note that , for otherwise would reduce to a constant. Now as and hence there is a positive number such that is bounded outside Moreover, is also bounded on being continuous on this compact subset of C and so  is bounded on C. If has no roots, then  is also entire and so, by Liouville’s Theorem,  is constant and therefore  is constant. This contradiction completes the proof of the theorem.

12.2 Other Applications

We can now easily prove a fascinating theorem about complex numbers.

Theorem 2.  Every non-constant complex polynomial of degree has exactly  zeros.

Proof. For every positive integer  and all complex numbers  we know that

Thus for where we get     

Multiplying by we get    

from which it follows that is divisible by .

Now, by the Fundamental Theorem of Algebra, there is a value such that So where is a polynomial of degree . By induction if  is a polynomial of degree , then     

where and  are constants. Consequently, every non-constant complex polynomial of degree  has exactly  zeros.

Theorem 3. Suppose that  is a polynomial with real coefficients. If  is a non-real zero of , then the conjugate of  is also a zero of .

Proof. Suppose that is a polynomial with real coefficients. Since  is a zero of By conjugation, Then and consequently, is a zero of .