Fundamental Theorem of Algebra


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To understand the fundamental theorem of algebra you need to know:

What are polynomials, how to find out the degree of a polynomial, and what are the roots of an equation.

The Fundamental Theorem of Algebra shows that the factors of the polynomials can be found from the roots and vice versa. For example, if you have a quadratic equation like

this equation can also be written as

- alpha and beta are the solutions of this equation.

The theorem itself is a bit more complex (and involves complex numbers!) and states that the polynomial of the n-th degree has n solutions, be they real or complex numbers.

So here is a polynomial of the form

P (x) =

this polynomial can also be written as

P (x) =

where are roots or solutions of the equation

You will not find the great need for this theorem until you get to the end of your GCSE or A level studies, however, it may be useful to know it anyway since you can immediately tell how many solutions an equation should have (although you may not necessarily know how to find them!).

Many famous mathematicians were interested in this theorem. Working on or around the solutions of equations other things became invented too, like the negative and complex numbers.

An Italian mathematician, Gerolamo Cardano (1501-1576) worked on finding solutions to equations and stumbled upon negative numbers. Before him mathematicians would just skip over that bit as they thought that the only possible numbers are positive (since you can never have negative things in nature). But Cardano describe the negative numbers as 'fictitious' and counted them as possible solutions to the equations.

Click here to find more about the history of negative numbers or if you are really into this, the history of complex numbers.

Some people that were interested in the Fundamental Theory of Algebra were Leibniz, Bernoulli and Euler, but the first attempt to prove the theorem was made by a French mathematician d'Alembert in 1746. The first textbook to describe the proof was Cours d'analyse de l'Ecole Royale Polytechnique, published in 1821 and written by Cauchy.



See the fronticepiece of the book in which the Fundamental Theorem of Algebra was first proved by clicking on the image below.

This book was published in Paris, by the Ecole Polytechnique - visit it here.

See some other famous places from the history of mathematics.

Or meet some famous mathematicians.

You can also climb a mathematical tree.


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