Operations with Polynomials - Ellen Moore`s 7010 Portfolio
... F- Multiply the first term in each binomial. O- Multiply the outside terms- first one in the first binomial and last one in the second. I- Multiply the inside terms which will be the last one in the first binomial and the first in the second one. L-Multiply the last term in each binomial. Then you w ...
... F- Multiply the first term in each binomial. O- Multiply the outside terms- first one in the first binomial and last one in the second. I- Multiply the inside terms which will be the last one in the first binomial and the first in the second one. L-Multiply the last term in each binomial. Then you w ...
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... Let (a, b, c) be a triple of complex numbers with c not belonging to the set of negative integers. For a complex number w and a non negative integer n, use Pochhammer symbol (w)n , to denote the expression : (w)n = w(w + 1) . . . (w + n − 1). The Gauss hypergeometric function, 2 F1 , is then defined ...
... Let (a, b, c) be a triple of complex numbers with c not belonging to the set of negative integers. For a complex number w and a non negative integer n, use Pochhammer symbol (w)n , to denote the expression : (w)n = w(w + 1) . . . (w + n − 1). The Gauss hypergeometric function, 2 F1 , is then defined ...
mathematics department 2003/2004
... 5. First and Second Fundamental Theorem of Calculus 6. integration by substitution / integration by parts 7. integration of rational functions 8. reduction formula 9. Improper integral 10. application to the finding of - plane area - arc length - volume of solids of revolution - area of surface of r ...
... 5. First and Second Fundamental Theorem of Calculus 6. integration by substitution / integration by parts 7. integration of rational functions 8. reduction formula 9. Improper integral 10. application to the finding of - plane area - arc length - volume of solids of revolution - area of surface of r ...
Chapter 11 Handout
... written as either a terminating decimal or a repeating decimal; eg 19/8=2.374; 1/3 =.3333... Irrational number: a number whose decimal form is nonterminating and nonrepeating eg √5 or pi√3 pi/2 11.5 Multiplying and dividing rational expressions To multiply: multiply numerators and denominators, then ...
... written as either a terminating decimal or a repeating decimal; eg 19/8=2.374; 1/3 =.3333... Irrational number: a number whose decimal form is nonterminating and nonrepeating eg √5 or pi√3 pi/2 11.5 Multiplying and dividing rational expressions To multiply: multiply numerators and denominators, then ...
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... An alternate statement of the problem is to ask for all triangular numbers which are squares. ...
... An alternate statement of the problem is to ask for all triangular numbers which are squares. ...
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... A complex number of the form a + bi, where a, b ∈ Z, is called a Gaussian integer. It is easy to see that the set S of all Gaussian integers is a subring of C; specifically, S is the smallest subring containing {1, i}, whence S = Z[i]. Z[i] is a Euclidean ring, hence a principal ring, hence a unique ...
... A complex number of the form a + bi, where a, b ∈ Z, is called a Gaussian integer. It is easy to see that the set S of all Gaussian integers is a subring of C; specifically, S is the smallest subring containing {1, i}, whence S = Z[i]. Z[i] is a Euclidean ring, hence a principal ring, hence a unique ...
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... Theorem (Nicomachus). The sum of the cubes of the first n integers is equal to the square of the nth triangular number. To put it algebraically, n X ...
... Theorem (Nicomachus). The sum of the cubes of the first n integers is equal to the square of the nth triangular number. To put it algebraically, n X ...
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... u is positive since -u is also a unit. Then either u = ot , or there is an n n+i ~n —n integer n such that ot < u < ot . Now ot is a unit, implying a u also is. But then 1-< oT u < a, which was shown impossible in the first part of the proof. Hence the onfy units in Q(\/5) a r e given in (3). We now ...
... u is positive since -u is also a unit. Then either u = ot , or there is an n n+i ~n —n integer n such that ot < u < ot . Now ot is a unit, implying a u also is. But then 1-< oT u < a, which was shown impossible in the first part of the proof. Hence the onfy units in Q(\/5) a r e given in (3). We now ...
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.