Chapter 10
... A quadratic equation you may recall from chapter 5 has the following form: ax2 + bx + c = 0 It is any equation, which is second degree (contains a term with a power of 2 and none that are higher). ...
... A quadratic equation you may recall from chapter 5 has the following form: ax2 + bx + c = 0 It is any equation, which is second degree (contains a term with a power of 2 and none that are higher). ...
MATH 123: ABSTRACT ALGEBRA II SOLUTION SET # 9 1. Chapter
... is always a prime or zero. If char = 0, this will be impossible, and if p = 2, no such field exists with char(F ) = 1. However, if p ≥ 3, as long as char(F ) divides p − 1, then f (x) has a multiple root. (NOTE: F need not be a finite field, consider Z/2Z(t) with char equal to 2.) Problem 4 Let α1 , ...
... is always a prime or zero. If char = 0, this will be impossible, and if p = 2, no such field exists with char(F ) = 1. However, if p ≥ 3, as long as char(F ) divides p − 1, then f (x) has a multiple root. (NOTE: F need not be a finite field, consider Z/2Z(t) with char equal to 2.) Problem 4 Let α1 , ...
Day
... I can use the exponent rules for multiplying and dividing monomials involving one and two variables with positive exponents. I can use the exponent rule for the power of a power to simplify expressions involving one and two variables with positive exponents. I can demonstrate and explain why t ...
... I can use the exponent rules for multiplying and dividing monomials involving one and two variables with positive exponents. I can use the exponent rule for the power of a power to simplify expressions involving one and two variables with positive exponents. I can demonstrate and explain why t ...
Document
... “decompose” it meaning break down into the fractions that were added together to get this answer ...
... “decompose” it meaning break down into the fractions that were added together to get this answer ...
Fast Fourier Transforms
... generality, was first used by Gauss around 1805 for calculating the periodic orbits of asteroids from a finite number of observations. In fact, Gauss’s recursive algorithm predates even Fourier’s introduction of harmonic analysis by two years. So, of course, the algorithm is universally called the C ...
... generality, was first used by Gauss around 1805 for calculating the periodic orbits of asteroids from a finite number of observations. In fact, Gauss’s recursive algorithm predates even Fourier’s introduction of harmonic analysis by two years. So, of course, the algorithm is universally called the C ...
PDF
... Claim we can assume, without loss of generality, that βi ∈ Z. For if not, take all the expressions formed by substituting for one or more of the βi one of its conjugates, and multiply those by the equation above. The result is a new expression of the same form (with different αi ), but where the co ...
... Claim we can assume, without loss of generality, that βi ∈ Z. For if not, take all the expressions formed by substituting for one or more of the βi one of its conjugates, and multiply those by the equation above. The result is a new expression of the same form (with different αi ), but where the co ...
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.