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In this section, we show that factoring over Q (the rational numbers) and over Z (the integers) is essentially the same problem.. The content of a polynomial p ∈ Z[X], denoted "cont(p)", is, up to its sign, the greatest common divisor of its coefficients.
In algebra, the polynomial remainder theorem or little Bézout's theorem (named after Étienne Bézout) [1] is an application of Euclidean division of polynomials.It states that, for every number , any polynomial is the sum of () and the product of and a polynomial in of degree one less than the degree of .
For example, the term 2x in x 2 + 2x + 1 is a linear term in a quadratic polynomial. The polynomial 0, which may be considered to have no terms at all, is called the zero polynomial . Unlike other constant polynomials, its degree is not zero.
Two problems where the factor theorem is commonly applied are those of factoring a polynomial and finding the roots of a polynomial equation; it is a direct consequence of the theorem that these problems are essentially equivalent.
For, if one applies Euclid's algorithm to the following polynomials [2] + + + and + +, the successive remainders of Euclid's algorithm are +, +,,. One sees that, despite the small degree and the small size of the coefficients of the input polynomials, one has to manipulate and simplify integer fractions of rather large size.
Degree: The maximum exponents among the monomials.; Factor: An expression being multiplied.; Linear factor: A factor of degree one.; Coefficient: An expression multiplying one of the monomials of the polynomial.
In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions that multiply two other subexpressions, at least one of which is an addition, by the equivalent sum of products, continuing until the expression becomes a ...
For example, if K is a field with q elements, then the polynomials 0 and X q − X both define the zero function. For every a in R , the evaluation at a , that is, the map P ↦ P ( a ) {\displaystyle P\mapsto P(a)} defines an algebra homomorphism from K [ X ] to R , which is the unique homomorphism from K [ X ] to R that fixes K , and maps X ...