The Fundamental Theorem of Algebra

Fundamental Theorem of Algebra:

Each and every branch of mathematics consists of its own basic theorem(s). When you check out basic in the dictionary, you will see that it associates to the foundation or the base or is elementary. The basic theorems are significant foundations for the rest of material to follow.

Some of the fundamental theorems or principles are below:

Fundamental Theorem of Arithmetic:

Each and every integer more than one is either prime or can be stated as a unique product of the prime numbers.

Fundamental Theorem of Linear Programming:

When there is a solution to linear programming problem, then it will take place at a corner point, or on the line segment among two corner points.

Fundamental Counting Principle:

When there are m ways to do one thing, and n ways to do the other, then there are m*n ways of doing the both.

Fundamental Theorem of Algebra:

Each and every polynomial in one variable of degree n>0 has at least one real and complex zero.

Some of the textbooks states at least on zero in complex number system. That is accurate. Though, most of the students forget that real are too complex numbers.

Corollary to the Fundamental Theorem of Algebra:

Each and every polynomial in one variable of degree n>0 has precisely n, not essentially distinct, real or complex zeros.

Linear Factorization Theorem:

It is a polynomial in one variable of degree n>0 and 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 function.

Note that each factor is a linear factor (that is, all x's are increased to the first power), however that there might be complex roots included.

The favored way of writing a polynomial is to utilize the linear and irreducible quadratic factorization. In this factorization, all the radicals are removed. This comprises complex numbers including i (keep in mind, i is sqrt(-1)).

There are two kinds of irreducible quadratic factors.

Irreducible over the Reals:

Whenever the quadratic factors contain no real roots, just complex roots including i, it is stated to be irreducible over the reals. This might include square roots, however not the square roots of negative numbers.

Irreducible over the Rationals:

Whenever the quadratic factors contain no rational roots, just irrational roots including radicals or complex numbers, then it is stated to be irreducible over the rationals. This is the favored form when the coefficients of polynomial are rational, or even better, integers.

Complex Roots:

The Complex roots come in pairs. This is why the utmost number of positive or negative real roots (that is, Descartes' Rule of Signs) should reduce by two. It cannot reduce by one as the only place for the roots to go is to the complex field, and they encompass to come in pairs. The other complex number that works is the complex conjugate.

Square Roots:

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

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