the fundamental theorem of algebra

Q: What is the Fundamental Theorem of Algebra?

A: The Fundamental Theorem of Algebra is every polynomial function has at least one zero in a complex number system.

Every nth-degree polynomial function has precisely n zeros. So if a polynomial's largest degree (n) is 3, it has exactly 3 zeros, or solutions. This idea was first proved by Carl Friedrich Gauss, a famous German mathematician who lived from 1777-1855.

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

 A related Theorem is the Linear Factorization Theorem.

Carl Friedrich Gauss 

Q: What is the Linear Factorization Theorem?

A: The Linear Factorization Theorem is a polynomial function with a degree of n, has a n  linear factors.

f(x) =  (x- )(x- ). . .(x- )   where  , , . . .,  are complex numbers.

These theorems are called extistence theorems because they do not describe how to find the zeros of a polynomial function.

For example:

f(x)=-6x+9 is a second degree polynomial and has exactly two zeros.  Solving for x, we find that the two zeros are -3 and -3. 

f(x)=+4x is a third degree polynomial and has exactly three zeros.  Solving for x, we find that the three zeros are 0, -2i, and 2i.

Sometimes polynomial functions have zeros that are not real such as the equation above (2i and -2i).  These zeros are called complex zeros.

Q: What are Complex Zeros?

A: Complex numbers, or imaginary numbers, are an extension of real numbers obtained by adjoining an imaginary unit, i. Every complex number can be written in the form a+bi or a-bi and they always come in pairs.

These kinds of imaginary numbers occur when you have to take the square root of a negative number.  Since you have to add a plus or minus before any answer that you get by taking a square root, this creates two answers.  That is why these complex numbers come in pairs.  In a graph, complex number solutions will not touch the x- axis. Therefore, if you are dealing with a third-degree polynomial, you know that it will have three zeros. The three zeros could be all real or two complex and one real. However, it would be impossible to have two real zeros and one complex.

For example:

Determine how many zeros this polynomial equation has by looking at it's graph and determine whether they are real or complex.

*To find all of the zeros in this graph, you must first identify what type of polynomial equation makes this kind of graph.

*The equation to this graph would start with an making it a fourth degree polynomial, meaning it has four real and/or complex number solutions.

*Now, look at the graph and determine how many x-intercepts there are.  This graph shows that there are two x-intercepts.

*Now that we know this polynomial has two real solutions, we can determine that it also has two imaginary solutions.  This would make a total of four solutions which matches up with the largest degree of the polynomial.

For more problems similar to this one, click here!

Q: Can I create a polynomial from a given list of zeros?

A:Yes!

For example:

Find all the zeros of (x) = x⁴ – 3x³ + 6x² + 2x-60

zeros given: 1+3i

Since 1+3i is a complex zeros and it comes with a partner, you know 1-3i is also a zero.

Multiply these two factors of f(x) to obtain x²-2x+10

Then divide using either long or synthetic division to  get x²-x-6=0

Factor to find the solutions x=3 and x=-2

The zeros are 1+3i, 1-3i, 3, and -2

It is also possible to reverse the operation and create the polynomial from the zeros by multiplying

(x-(1+3i))(x-(1-3i))(x+2)(x-3) to get the original polynomial function.

For further explanations and examples, click here!