Finding Factors of Factor Rings over the Gaussian Integers
... arithmetic in Z[i]; first, by defining primes and illustrating which Gaussian integers are prime, and then by proving the existence of unique factorization into these primes. (This can now be recognized as the beginning of algebraic number theory, which seeks to do for Z[ξ ] = {a + bξ + cξ 2 + · · · ...
... arithmetic in Z[i]; first, by defining primes and illustrating which Gaussian integers are prime, and then by proving the existence of unique factorization into these primes. (This can now be recognized as the beginning of algebraic number theory, which seeks to do for Z[ξ ] = {a + bξ + cξ 2 + · · · ...
College Algebra Definitions and Procedures
... If p/q is a zero of P(x), then p is a factor of a0 and q is a factor of an. Descartes’ Rule of Signs Let P(x) be a polynomial function with real coefficients and a nonzero constant term. The number of positive real zeros of P(x) is either: 1. The same as the number of variations of sign in P(x) 2. L ...
... If p/q is a zero of P(x), then p is a factor of a0 and q is a factor of an. Descartes’ Rule of Signs Let P(x) be a polynomial function with real coefficients and a nonzero constant term. The number of positive real zeros of P(x) is either: 1. The same as the number of variations of sign in P(x) 2. L ...
Sample Scope and Sequence for Algebra II for the Common Core
... 1. Make sense of problems polynomials.) and persevere in solving A-‐APR.2 (Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the them. remainder on division by x – a is p( ...
... 1. Make sense of problems polynomials.) and persevere in solving A-‐APR.2 (Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the them. remainder on division by x – a is p( ...
Solution
... (a) Note that the original form of this problem (where R is not a field) is not correct at all. For example, over Z/6 the polynomial 5x + 1 is not irreducible: (5x + 1) = (2x + 1)(3x + 1). Thus the situation over rings is much more complicated. However, over a field degrees add when polynomials are ...
... (a) Note that the original form of this problem (where R is not a field) is not correct at all. For example, over Z/6 the polynomial 5x + 1 is not irreducible: (5x + 1) = (2x + 1)(3x + 1). Thus the situation over rings is much more complicated. However, over a field degrees add when polynomials are ...
22-Factoring - Rose
... So ~1/115 of odd 100-digit numbers are prime Can start with a random large odd number and iterate, applying M-R to remove composites. We’ll soon find one that is a likely prime. Can repeat with different bases to improve probability that it’s prime. Maple’s nextprime() appears to do this, but also r ...
... So ~1/115 of odd 100-digit numbers are prime Can start with a random large odd number and iterate, applying M-R to remove composites. We’ll soon find one that is a likely prime. Can repeat with different bases to improve probability that it’s prime. Maple’s nextprime() appears to do this, but also r ...
Factorization
In mathematics, factorization (also factorisation in some forms of British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5, and the polynomial x2 − 4 factors as (x − 2)(x + 2). In all cases, a product of simpler objects is obtained.The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, or polynomials to irreducible polynomials. Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viète's formulas relate the coefficients of a polynomial to its roots.The opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms.Integer factorization for large integers appears to be a difficult problem. There is no known method to carry it out quickly. Its complexity is the basis of the assumed security of some public key cryptography algorithms, such as RSA.A matrix can also be factorized into a product of matrices of special types, for an application in which that form is convenient. One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types: QR decomposition, LQ, QL, RQ, RZ.Another example is the factorization of a function as the composition of other functions having certain properties; for example, every function can be viewed as the composition of a surjective function with an injective function. This situation is generalized by factorization systems.