Fundamental Theorem of Algebra Fundamental Theorem of Algebra The Fundamental Theorem of Algebra is one of the important theorems of the mathematics. The statement of the fundamental theorem of algebra is that every polynomial which has only one variable which is not constant and which has the coefficients which are complex possesses at least one root which is complex because we already know that the coefficients which are real and the roots which are real come inside the definition of numbers which are complex. According to the statement of the fundamental theorem of algebra given in the last paragraph, the fundamental theorem of algebra can alternatively be stated as follows. The fundamental theorem of algebra says that the field of the numbers which are complex is shut down algebraically. Other than the statements of the fundamental theorem of algebra given in the last 2 paragraphs, the fundamental theorem of algebra can also be stated in a different manner as follows. The fundamental theorem of algebra states that each single polynomial which has only single variable and is other than zero and which has coefficients which are complex possesses the

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no. of roots which are complex equal to the degree of the polynomial with the condition that every single root is seen up to the multiplicity of the root. After reading this statement of the fundamental theorem of algebra it seems that this is the more powerful statement of the fundamental theorem of algebra but it should be known that this statement of the fundamental theorem of algebra is just derived from the other statements of the theorem given in the earlier paragraphs by the way of the continuous division of the polynomial by the factors which are linear. Although the word algebra is used in the name of the theorem as the name â&#x20AC;&#x2DC;fundamental theorem of algebra â&#x20AC;&#x2DC; suggests but it should be known that there exists no proof of the fundamental theorem of algebra which is completely algebraic. Also the name of the theorem says that it is the fundamental theorem but the fundamental theorem of algebra is not at all fundamental for the modern type algebra because the name of the theorem was proposed at some time during which the study of the algebra was just related to the solutions containing polynomial having the coefficients which could be either complex or real. We have discussed enough about the various statements of the fundamental theorem of algebra so let us now discuss something about the proofs of the fundamental theorem of algebra. Every proof of the fundamental theorem of algebra contain some kind of the analysis and if not that then at least it will contain the concept of the continuity of the real functions or the complex functions which will be topological. Some of the proofs of this fundamental theorem of algebra also utilize such type of the functions which can be analytic or differential. These are the facts which have proved that the fundamental theorem of algebra is not at all fundamental and also that it is not even the theorem of the algebra.

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This because the proofs which are algebraic use only 2 things about the real numbers that each polynomial which has odd degree and contain coefficients which are real possesses real root and each positive real number possesses a square root. Example 1:- Factorise completely: f (x) = x4 â&#x20AC;&#x201C; 1 using fundamental theorem of algebra Solution:- We know that since n = 4, there are exactly 4 complex zeros, roots, and linear factors for f. The factorization for f could be done in this way: f (x) = x4 - 1 = (x2 - 1 ) (x2 + 1) = (x + 1)(x - 1)(x + i )(x - i) These are the four linear factors of f and the four zeros of f are x = Âą 1 and x = Âąi Example 2 : - Factoring a polynomial completely: f (x) = x3 - x2 using fundamental theorem of algebra. Solution :- The factorization for f could be done in this way, f (x) = x3 - x2 We can pull out common terms x2 : x3 - x2 = x2 (x - 1). = x2 ( x - 1 ) We have a factored the polynomial into three linear factors, thus the factorization is complete using fundamental theorem of algebra.

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Fundamental Theorem of Algebra
Fundamental Theorem of Algebra

Other than the statements of the fundamental theorem of algebra given in the last 2 paragraphs, the fundamental theorem of algebra can also...