The Fundamental Theorem of Algebra

The fundamental theorem of algebra states that a polynomial of degree n has exactly n zeros.

This includes complex numbers so some zeros are represented as crossing the x-axis, like this one.

 

set the equation equal to zero to find the x intercepts

 

subtract 5 from each side

 

 



 

 

so the factors are complex

 

 

 



 

 

 

The parabola shown on the graph will never cross the x-axis, but the zeros still exist. They are just imaginary. This also applies to functions of higher degrees.

 

 

 

The graph shown above crosses the x-axis twice, at 1 and -1, but i, and -i are also linear factors of the function. Therefore, the complex factors of the equation can be thought of as places where the function could potentially have crossed the x-axis.

 

Both of these examples have complex numbers as a part of the linear factorization of the function. Whenever there is one complex zero in the function, its conjugate must also be a factor.

 

So if x+4i is a factor in any polynomial, x-4i will also have to be present



 

 

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