3.4 - Fundamental Theorem of Algebra

Each branch of mathematics has its own fundamental theorem(s). If you check out fundamental in the dictionary, you will see that it relates to the foundation or the base or is elementary. Fundamental theorems are important foundations for the rest of the material to follow.

Here are some of the fundamental theorems or principles that occur in your text.

Fundamental Theorem of Arithmetic (pg 9)
Every integer greater than one is either prime or can be expressed as an unique product of prime numbers.
Fundamental Theorem of Linear Programming (pg 440)
If there is a solution to a linear programming problem, then it will occur at a corner point, or on a line segment between two corner points.
Fundamental Counting Principle (pg 574)
If there are m ways to do one thing, and n ways to do another, then there are m*n ways of doing both.

Fundamental Theorem of Algebra

Every polynomial in one variable of degree n>0 has at least one real or complex zero.

Now, your textbook says at least on zero in the complex number system. That is correct. However, most students forget that reals are also complex numbers, so I will try to spell out real or complex to make things simpler for you.

Corollary to the Fundamental Theorem of Algebra

Every polynomial in one variable of degree n>0 has exactly n, not necessarily distinct, real or complex zeros.

Linear Factorization Theorem

A polynomial in one variable of degree n>0 can be factored as

f(x)=an (x-c1) (x-c2) (x-c3) ... (x-cn)

Where an is the leading coefficient and each c1 ... cn is a real or complex root of the function.

Notice that each factor is a linear factor (all x's are raised to the first power), but that there may be complex roots involved.

Linear and Irreducible Quadratic Factorization

The preferred way of writing a polynomial is to use the linear and irreducible quadratic factorization. In this factorization, all radicals are eliminated. This includes complex numbers involving i (remember, i is the sqrt(-1)).

There are two types of irreducible quadratic factors. The book will be talking about irreducible over the reals. I prefer irreducible over the rationals (if the function has integer coefficients).

Irreducible over the Reals
When the quadratic factors have no real roots, only complex roots involving i, it is said to be irreducible over the reals. This may involve square roots, but not the square roots of negative numbers.
Irreducible over the Rationals
When the quadratic factors have no rational roots, only irrational roots involving radicals or complex numbers, then it is said to be irreducible over the rationals. This is the preferred form when the coefficients of the polynomial are rational, or even better, integers.

Complex Roots

Complex roots come in pairs. This is why the maximum number of positive or negative real roots (Descartes' Rule of Signs) must decrease by two. It can't decrease by one because the only place for the roots to go is into the complex field, and they have to come in pairs. The other complex number which works is the complex conjugate.

Square Roots

Square roots come in pairs. This is not necessarily true of other roots. The other square root which works is the conjugate of the first.