Solutions to selected problems from Chapter 2
... 2.10 The polynomial a(x) = x5 + x3 + 1 is irreducible over GF (2) if it is not divisible by any polynomial p(x) over GF (2) of degree 1 ≤ deg(p(x)) ≤ 2 = bdeg(a(x))/2c. Thus, we have to check if there exists a polynomial of the form p(x) = p2 x2 + p1 x + p0 over GF (2), which divides a(x). (a) As a( ...
... 2.10 The polynomial a(x) = x5 + x3 + 1 is irreducible over GF (2) if it is not divisible by any polynomial p(x) over GF (2) of degree 1 ≤ deg(p(x)) ≤ 2 = bdeg(a(x))/2c. Thus, we have to check if there exists a polynomial of the form p(x) = p2 x2 + p1 x + p0 over GF (2), which divides a(x). (a) As a( ...
Section 2
... A quadratic equation in x is an equation that can be written in the general form. ax 2 bx c 0 where a, b, and c are real numbers, with a 0 . It is also called a second-degree polynomial equation in x. *Solving Quadratic Equation by Factoring The Zero-Product Principle If AB 0 , then A 0 ...
... A quadratic equation in x is an equation that can be written in the general form. ax 2 bx c 0 where a, b, and c are real numbers, with a 0 . It is also called a second-degree polynomial equation in x. *Solving Quadratic Equation by Factoring The Zero-Product Principle If AB 0 , then A 0 ...
Math 1302- Short Quiz - Angelo State University
... problem at the very beginning – save those for the end of class. 1. Number of absences up to today. ___________ Give me a close approximation if not sure of exact number. A blank answer will result in points counted off. ...
... problem at the very beginning – save those for the end of class. 1. Number of absences up to today. ___________ Give me a close approximation if not sure of exact number. A blank answer will result in points counted off. ...
Full text
... And rational functions are closed under the Hadamard product! (See [1], p. 85.) The larger (any maybe even more Important) class of holonomic functions (solutions of linear differential equations with polynomial coefficients) is also closed under the Hadamard product. Their Taylor coefficients fulfi ...
... And rational functions are closed under the Hadamard product! (See [1], p. 85.) The larger (any maybe even more Important) class of holonomic functions (solutions of linear differential equations with polynomial coefficients) is also closed under the Hadamard product. Their Taylor coefficients fulfi ...
College Algebra – Chapter 3 “Are You Ready” Review Name: 1
... The remainder theorem tells us what the remainder WOULD be IF we were to divide a polynomial by a linear factor. To use it, you plug the zero into the function and evaluate it. In other words, if f (c) = r, then r will be the remainder if you divide f (x) by (x – c). We can use this to determine whi ...
... The remainder theorem tells us what the remainder WOULD be IF we were to divide a polynomial by a linear factor. To use it, you plug the zero into the function and evaluate it. In other words, if f (c) = r, then r will be the remainder if you divide f (x) by (x – c). We can use this to determine whi ...
Automatic Geometric Theorem Proving: Turning Euclidean
... It is√enough to show that the generators of T (for this example, t) are elements of H. For this example, it happens to turn out that t ∈ H ⊆ ...
... It is√enough to show that the generators of T (for this example, t) are elements of H. For this example, it happens to turn out that t ∈ H ⊆ ...
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.