Ch8 - ClausenTech
... 8-5 The Remainder and the Factor Theorems
Remainder Theorem: If P(x) is a polynomial of degree n (n>0), then for any number r,
P(x)= Q(x) (x-r)+P(r) , where Q(x), is a polynomial of degree n-1. For the polynomial
P(x), the function value P(r) is the remainder when P(x) is divided by x-r.
Factor Theo ...
25. Abel`s Impossibility Theorem
... Theory. Needless to say, the proof cannot readily be condensed into several pages without
significant background. Dörrie arrives at the following theorem from Kronecker (from 1856
in the Monatsberichte der Berliner Akademie):
Theorem. An irreducible polynomial equation of odd prime degree which is s ...
Academy Algebra II 5.7: Apply the Fundamental Theorem of Algebra
... • The number of positive real zeros of f is
equal to the number of sign changes in
the sign of the coefficients of f(x) or is
less than this by an even number.
• The number of negative real zeros of f is
equal to the number of changes in sign of
the coefficients of f(-x) or is less than this
by an e ...
... is a polynomial with positive degree, i.e. n > 1 and an 6= 0 such that T (W ) is a
zero function. It follows from the Bezout’s theorem that W has at most n roots
(in fact this is true over any integral domain). Thus since k is an infinite field,
then there exists a ∈ k which is not a root of W . In ...
Fundamental theorem of algebra
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with an imaginary part equal to zero.Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed.The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n roots. The equivalence of the two statements can be proven through the use of successive polynomial division.In spite of its name, there is no purely algebraic proof of the theorem, since any proof must use the completeness of the reals (or some other equivalent formulation of completeness), which is not an algebraic concept. Additionally, it is not fundamental for modern algebra; its name was given at a time when the study of algebra was mainly concerned with the solutions of polynomial equations with real or complex coefficients.