Math 1
... If n is odd, then a has one real nth root n a If n is even and a > 0, then a has two real nth roots n a If n is even and a = 0, then a has one real nth root n 0 If n is even and a < 0, then a has no real nth roots ...
... If n is odd, then a has one real nth root n a If n is even and a > 0, then a has two real nth roots n a If n is even and a = 0, then a has one real nth root n 0 If n is even and a < 0, then a has no real nth roots ...
Document
... First, check for a GCF. Then, since the polynomial has two terms, check for the difference of squares. 6x2 – 96 = 6(x2 – 16) = 6(x2 – 42) = 6(x + 4)(x – 4) Answer: 6(x + 4)(x – 4) ...
... First, check for a GCF. Then, since the polynomial has two terms, check for the difference of squares. 6x2 – 96 = 6(x2 – 16) = 6(x2 – 42) = 6(x + 4)(x – 4) Answer: 6(x + 4)(x – 4) ...
Full text
... The study of Bernoulli, Euler, and Eulerian polynomials has contributed much to our knowledge of the theory of numbers. These polynomials are of basic importance in several parts of analysis and calculus of finite differences , and have applications in various fields such as statistics, numerical an ...
... The study of Bernoulli, Euler, and Eulerian polynomials has contributed much to our knowledge of the theory of numbers. These polynomials are of basic importance in several parts of analysis and calculus of finite differences , and have applications in various fields such as statistics, numerical an ...
slides - faculty.ucmerced.edu
... Proof: Use a proof by contraposition. Suppose x and y are not both even. Then, one or both are odd. Without loss of generality, assume that x is odd. Then x = 2m + 1 for some integer k. Case 1: y is even. Then y = 2n for some integer n, so x + y = (2m + 1) + 2n = 2(m + n) + 1 is odd. Case 2: y is od ...
... Proof: Use a proof by contraposition. Suppose x and y are not both even. Then, one or both are odd. Without loss of generality, assume that x is odd. Then x = 2m + 1 for some integer k. Case 1: y is even. Then y = 2n for some integer n, so x + y = (2m + 1) + 2n = 2(m + n) + 1 is odd. Case 2: y is od ...
Other-Equations
... Quadratic-Form Equations Equations that may be written in the form ax2N + bxN + c = 0 are known as quadratic form equations. For example, the equation 2x8 + 4x4 + 2 = 0 is a quadratic-form equation since the power of the first term, 8, is twice the power of the second term 4. The third term is a re ...
... Quadratic-Form Equations Equations that may be written in the form ax2N + bxN + c = 0 are known as quadratic form equations. For example, the equation 2x8 + 4x4 + 2 = 0 is a quadratic-form equation since the power of the first term, 8, is twice the power of the second term 4. The third term is a re ...
Chapter 1
... 5.2.3.2. Definition of Integer Division – For all integers a, b, and c, b 0, a b = c if and only if c x b = a 5.2.3.3. Procedure for Dividing Integers 5.2.3.3.1. Dividing two positive integers: Divide digits, keep the sign (+) 5.2.3.3.2. Dividing two negative integers: Divide digits, change the ...
... 5.2.3.2. Definition of Integer Division – For all integers a, b, and c, b 0, a b = c if and only if c x b = a 5.2.3.3. Procedure for Dividing Integers 5.2.3.3.1. Dividing two positive integers: Divide digits, keep the sign (+) 5.2.3.3.2. Dividing two negative integers: Divide digits, change the ...
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.