![18. [Multiples / Factors / Primes]](http://s1.studyres.com/store/data/013189166_1-74f5d0f8cc5c70ec77f92ed53cb96d29-300x300.png)
Rational values of the arccosine function
... For even n, a different method is suggested in that book, distinguishing between the cases√n = 2j and n not a power of 2. Thus, it is obtained that (1/π) arccos(1/ n) is rational if and only if n ∈ {1, 2, 4}. The relation (2) can be read in term of Chebyshev polynomials of the first kind. These poly ...
... For even n, a different method is suggested in that book, distinguishing between the cases√n = 2j and n not a power of 2. Thus, it is obtained that (1/π) arccos(1/ n) is rational if and only if n ∈ {1, 2, 4}. The relation (2) can be read in term of Chebyshev polynomials of the first kind. These poly ...
Elementary Algebra with Applications
... You will demonstrate your competence: 11.a. by completing assignments, quizzes, lab work, and tests with a satisfactory score as detailed in the grading scale 11.b. ...
... You will demonstrate your competence: 11.a. by completing assignments, quizzes, lab work, and tests with a satisfactory score as detailed in the grading scale 11.b. ...
(pdf)
... is the product of two fields. But rewriting Z[i] as Z[X]/(X 2 + 1) we see that Z[i]/(p) = Fp [X]/(X 2 +1) is a product of fields if and only if X 2 +1 has a root (mod p). Now (Fp )× ∼ = Z/(p − 1)Z, so −1 is a square in Fp if and only if −1 ≡ (p − 1)/2 (mod p − 1) is even. But this occurs if and only ...
... is the product of two fields. But rewriting Z[i] as Z[X]/(X 2 + 1) we see that Z[i]/(p) = Fp [X]/(X 2 +1) is a product of fields if and only if X 2 +1 has a root (mod p). Now (Fp )× ∼ = Z/(p − 1)Z, so −1 is a square in Fp if and only if −1 ≡ (p − 1)/2 (mod p − 1) is even. But this occurs if and only ...
![18. [Multiples / Factors / Primes]](http://s1.studyres.com/store/data/012793609_1-c9c531d443fb23eb7f01ba7a838248e1-300x300.png)
37(2)
... Also, it is clear that G is bijective and x < G(x) for all x > 0. Thus, G~l exists and is also strictly increasing with G~l(x) < x. Let u = G_1(x). Then G(u) = x and u = 3x + v8x 2 +1. Since u < x, we have u = 3x - V8x2 4-1. Also, since 8(G_1(x))2 +1 = (8x - 3V8x2 +1) 2 is a perfect square, it follo ...
... Also, it is clear that G is bijective and x < G(x) for all x > 0. Thus, G~l exists and is also strictly increasing with G~l(x) < x. Let u = G_1(x). Then G(u) = x and u = 3x + v8x 2 +1. Since u < x, we have u = 3x - V8x2 4-1. Also, since 8(G_1(x))2 +1 = (8x - 3V8x2 +1) 2 is a perfect square, it follo ...
Precal CPA summer Assignment
... http://mathbits.com/MathBits/TeacherResources/Algebra2/Algebra2.htm ...
... http://mathbits.com/MathBits/TeacherResources/Algebra2/Algebra2.htm ...
Name - West Essex Regional School District
... http://mathbits.com/MathBits/TeacherResources/Algebra2/Algebra2.htm ...
... http://mathbits.com/MathBits/TeacherResources/Algebra2/Algebra2.htm ...
Rational Expressions
... When we have a common denominator, as with fractions, we simply add or subtract the numerators. We do have to be cautious because when subtracting it is the entire numerator that's subtracted, so we must use the distributive property to subtract. Concept Example: 7 3 = 4 ...
... When we have a common denominator, as with fractions, we simply add or subtract the numerators. We do have to be cautious because when subtracting it is the entire numerator that's subtracted, so we must use the distributive property to subtract. Concept Example: 7 3 = 4 ...
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.