Editing Fundamental theorem of algebra (section)

==Equivalent statements==
There are several equivalent formulations of the theorem:
* ''Every [[univariate polynomial]] of positive degree with real coefficients has at least one complex [[zero of a function|root]].''
* ''Every univariate polynomial of positive degree with complex coefficients has at least one complex [[zero of a function|root]].''
*:This implies immediately the previous assertion, as real numbers are also complex numbers. The converse results from the fact that one gets a polynomial with real coefficients by taking the product of a polynomial and its [[complex conjugate]] (obtained by replacing each coefficient with its complex conjugate). A root of this product is either a root of the given polynomial, or of its conjugate; in the latter case, the conjugate of this root is a root of the given polynomial. 
* ''Every univariate polynomial of positive degree {{mvar|n}} with complex coefficients can be [[factorization|factorized]] as <math display =block>c(x-r_1)\cdots(x-r_n),</math> where <math>c, r_1, \ldots, r_n</math> are complex numbers.''
*:The {{mvar|n}} complex numbers <math>r_1, \ldots, r_n</math> are the roots of the polynomial. If a root appears in several factors, it is a [[multiple root]], and the number of its occurrences is, by definition, the [[multiplicity (mathematics)|multiplicity]] of the root.
*: The proof that this statement results from the previous ones is done by [[recursion]] on {{mvar|n}}: when a root <math>r_1</math> has been found, the [[polynomial division]] by <math>x-r_1</math> provides a polynomial of degree <math>n-1</math> whose roots are the other roots of the given polynomial.
The next two statements are equivalent to the previous ones, although they do not involve any nonreal complex number. These statements can be proved from previous factorizations by remarking that, if {{mvar|r}} is a non-real root of a polynomial with real coefficients, its complex conjugate <math>\overline r</math> is also a root, and <math>(x-r)(x-\overline r)</math> is a polynomial of degree two with real coefficients (this is the [[complex conjugate root theorem]]). Conversely, if one has a factor of degree two, the [[quadratic formula]] gives a root.
* ''Every univariate polynomial with real coefficients of degree larger than two has a factor of degree two with real coefficients.''
* ''Every univariate polynomial with real coefficients of positive degree can be factored as <math display = block>cp_1\cdots p_k,</math> where {{mvar|c}} is a real number and each <math>p_i</math> is a [[monic polynomial]] of degree at most two with real coefficients. Moreover, one can suppose that the factors of degree two do not have any real root.''