The FUNdamental Theorem of Algebra

In this section we will discuss the Fundamental Theorem of Algebra, the Linear Factorization Theorem,  Complex Zeros, and much more.

Definitions

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

Linear Factorization Theorem: If f(x) is a polynomial of a degree n where n > 0, f has precisely n linear factors. f(x) = f(x)=a_n(x-c₁)(x-c₂)…(x-c_n) where c_1, c_{2, .....} c_{n ,  are complex numbers.}

_{Application and Examples}

_{To find the zeros of an equation one must first understand what a zero is. This is a point on the graph of any function where the line of the function crosses the x-axis. A zero can be found by plugging 0 in for the y value of a function and then solving for x.} 

Example:

f(x) = x-2

+2      +2

x = 2

For this equation the graph will cross the x-axis at the point (2,0)

Most functions will not be as easy to solve as the previous example. For these it is recommended that for smaller functions such as cubic or quadratic that one attempts factoring or the quadratic equation first.

Factoring Example:

x²-6x+9=f(x)

(x-3)(x-3)=f(x)

This equation has two zeros x=3 and x=3.

For larger functions containing exponents much larger than these shown a useful method to use would be synthetic division, which was demonstrated in an early section.

The zeros will not always be easy real numbers. Many equations will contain imaginary numbers for zeros. Before we discuss complex numbers we have to know what they are.

Imaginary Numbers- This is a sub category of complex numbers which includes numbers that cannot be expressed with out the use of non-real terms such as i. Examples: 3+i, i, 3-i

Equations that involve imaginary numbers will appear as though the whole graph has been shifted up or down along the x-axis. For these types of graphs imagine sliding the x-axis upwards and see if there is a point at which the is the same amount of x-intercepts as there should be based upon the highest exponent value.

When there are complex numbers as zero which involve in some way an imaginary number their conjugate will also be a zero. If a+bi, where a and b are real numbers, is a zero, than a-bi will also be a zero.

Never will there be one of the values for a conjugate pair be a zero and the other not be a zero if the equation is truly a function.

When Imaginary numbers are found in a graph they will appear in some form of this:

Real zeros on the other hand will appear as intercepts such as these: 

A graph with both real and non-real zeros may appear like this:

Working From Zeros Back to the Original Equation can be a long tedious process though never should it be too difficult.

Example: the zeros for a function are 2,3, and 7. Write the equation for this function as a polynomial.

(x-3)(x-2)(x-7)=f(x)

(x²-5x+6)(x-7)=f(x)

x³-12x²+41x-42=f(x)

To solve problems like this simply take the zeros and place them as you would find them at the end of factoring a polynomial. Then work backward as shown and FOIL the linear functions until they are all one large polynomial function again.