Pollard's p - 1 Method
... 1) Choose an integer k which is a multiple of most (or of all) of the numbers b ≤ B; e.g. k = B!. 2) Choose a (random) number a with 2 ≤ a ≤ n−2. 3) Compute r = rem(ak , n) by the power-mod method. 4) Compute d = (r − 1, n) by the Euclidean algorithm. 5) If d = 1 or d = n, start over (new a or new k ...
... 1) Choose an integer k which is a multiple of most (or of all) of the numbers b ≤ B; e.g. k = B!. 2) Choose a (random) number a with 2 ≤ a ≤ n−2. 3) Compute r = rem(ak , n) by the power-mod method. 4) Compute d = (r − 1, n) by the Euclidean algorithm. 5) If d = 1 or d = n, start over (new a or new k ...
math 1314 noes 3.3 and 3.4
... x3 2 x 2 5 x 6 x 2 x 2 4 x 3 0 Set each factor equal to zero and solve: The solutions (the rational zeros) are 3, 1, 2. So far, we’ve found the zeros of a polynomial whose highest power is 3. If the polynomial is of degree 4 or higher, we should try to find more than one fa ...
... x3 2 x 2 5 x 6 x 2 x 2 4 x 3 0 Set each factor equal to zero and solve: The solutions (the rational zeros) are 3, 1, 2. So far, we’ve found the zeros of a polynomial whose highest power is 3. If the polynomial is of degree 4 or higher, we should try to find more than one fa ...
07 some irreducible polynomials
... [1.0.6] Example: P (x) = x6 + x5 + x4 + x3 + x2 + x + 1 is irreducible over k = Z/p for prime p = 3 mod 7 or p = 5 mod 7. Note that x7 − 1 = (x − 1)(x6 + x5 + x4 + x3 + x2 + x + 1) Thus, any root of P (x) = 0 has order 7 or 1 (in whatever field it lies). The only element of order 1 is the identity e ...
... [1.0.6] Example: P (x) = x6 + x5 + x4 + x3 + x2 + x + 1 is irreducible over k = Z/p for prime p = 3 mod 7 or p = 5 mod 7. Note that x7 − 1 = (x − 1)(x6 + x5 + x4 + x3 + x2 + x + 1) Thus, any root of P (x) = 0 has order 7 or 1 (in whatever field it lies). The only element of order 1 is the identity e ...
2/15/10 (Monday) NOTES CLASSWORK HOMEWORK No School
... (Factor each number under the radical sign and find the square root of a perfect square and bring that number outside. Keep the non-perfect square factor under the radical sign.) ...
... (Factor each number under the radical sign and find the square root of a perfect square and bring that number outside. Keep the non-perfect square factor under the radical sign.) ...
Function Operations
... The notation used for the composition of functions looks like this: (f g)(x). The composition of the function f with g is defined as follows: (f g)(x) = f(g(x)), notice that in this case, the function g is inside of the function f To find the composition of two functions: Step 1: Rewrite the com ...
... The notation used for the composition of functions looks like this: (f g)(x). The composition of the function f with g is defined as follows: (f g)(x) = f(g(x)), notice that in this case, the function g is inside of the function f To find the composition of two functions: Step 1: Rewrite the com ...
Algebra 1A Study Guide for Final Exam - coachpate
... binomial and trinomial and identifying them. Make sure you can find the degree of a polynomial and can list one in descending or ascending order by variable. Plus all operations on polynomials (add, subtract, multiply, FOIL method, and special products—these will save you time). Sections—All section ...
... binomial and trinomial and identifying them. Make sure you can find the degree of a polynomial and can list one in descending or ascending order by variable. Plus all operations on polynomials (add, subtract, multiply, FOIL method, and special products—these will save you time). Sections—All section ...
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.