2.5 The Fundamental Theorem of Algebra 

The fundamental theorem of algebra states that a polynomial function of a degree n has exactly n zeroes. This is true if n>0. These zeroes can either be real or non-real numbers.

So, whether real or non-real, the function  has exactly 2 zeroes, the function  has exactly 3 zeroes, and so on.


Real Zeroes of Polynomial Functions

• Some functions have zeroes that are all real 

Consider the function:


when factored, the function simplifies to

(x+4)(x+2)

so, the resulting zeroes of this function are -4 and -2. This function has exactly 2 real zeroes. 

Non-Real Zeroes of Polynomial Functions

Although it is nice when when all of the zeroes of a function are real, this is often not the case. Many functions have non-real zeroes. 

• Non-real zeroes are written in the form a+bi
• Non-real zeroes ALWAYS COME IN PAIRS!!! This means that any given function can only have an EVEN number of non-real zeroes 
• The pairs that non-real zeroes come in are the non-real zero and its conjugate

A function  has 4 real zeroes 

when factored, it comes out to be: 
     
           

NOTICE: The non real zero  and its conjugate  - are both zeroes

Graphing Polynomial Functions with Non-Real Zeroes 

From this illustration, it is demonstrated that non-real zeros are zeros that DO NOT touch or cross the x axis on the graph of a function. 

In the graph of the function of degree 4 with no real zeros, it does not touch or cross the x axis. This means that the function still does have exactly 4 zeros, but they are all non-real

On the graph of the function with the degree of four with four real zeros, it crosses the x axis 4 times, at each point the graph crosses the x axis, this a real zero of the function

On the graph of the function with a degree of 6, it is demonstrated that this function has both real and non-real zeros. The points where x changes direction but does not touch the x axis represent the non-real zeros of the function. The two points where the graph crosses the x axis represent the real zeros of the function.

TRUE or FALSE

An equation with zero  must also have a zero at . TRUE 

• This statement is true because non-real zeros of a polynomial function always come in pairs 

It is possible for the equation  to only have 3 zeros. FALSE

• This statement is false because the fundamental theorem of algebra states that a polynomial to the nth degree must have exactly n zeros