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Ruffini's Rule
In mathematics, Ruffini's rule is a method for computation of the Euclidean division of a polynomial by a binomial of the form ''x – r''. It was described by Paolo Ruffini in 1809. The rule is a special case of synthetic division in which the divisor is a linear factor. Algorithm The rule establishes a method for dividing the polynomial: :P(x)=a_nx^n+a_x^+\cdots+a_1x+a_0 by the binomial: :Q(x)=x-r to obtain the quotient polynomial: :R(x)=b_x^+b_x^+\cdots+b_1x+b_0. The algorithm is in fact the long division of ''P''(''x'') by ''Q''(''x''). To divide ''P''(''x'') by ''Q''(''x''): # Take the coefficients of ''P''(''x'') and write them down in order. Then, write ''r'' at the bottom-left edge just over the line: #: \begin & a_n & a_ & \dots & a_1 & a_0\\ r & & & & & \\ \hline & & & & & \\ \end # Pass the leftmost coefficient (''a''''n'') to the bottom just under the line. #: \begin & a_n & a_ & \dots & a_1 & a_0\\ r & & & & & \\ \hline & a_n & & & & \\ & =b_ & & & & \end ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Remainder Theorem
In algebra, the polynomial remainder theorem or little Bézout's theorem (named after Étienne Bézout) is an application of Euclidean division of polynomials. It states that, for every number r, any polynomial f(x) is the sum of f(r) and the product of x-r and a polynomial in x of degree one less than the degree of f. In particular, f(r) is the remainder of the Euclidean division of f(x) by x-r, and x-r is a divisor of f(x) if and only if f(r)=0, a property known as the factor theorem. Examples Example 1 Let f(x) = x^3 - 12x^2 - 42. Polynomial division of f(x) by (x-3) gives the quotient x^2 - 9x - 27 and the remainder -123. By the polynomial remainder theorem, f(3)=-123. Example 2 Proof that the polynomial remainder theorem holds for an arbitrary second degree polynomial f(x) = ax^2 + bx + c by using algebraic manipulation: \begin f(x)-f(r) &= ax^2+bx+c-(ar^2+br+c)\\ &= a(x^2-r^2)+ b(x-r)\\ &= a(x-r)(x+r)+b(x-r)\\ &= (x-r)(ax +ar+ b) \end So, f(x) = (x - r)(ax + ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomials
In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; and they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Horner's Method
In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Horner himself, and can be traced back many hundreds of years to Chinese and Persian mathematicians. After the introduction of computers, this algorithm became fundamental for computing efficiently with polynomials. The algorithm is based on Horner's rule, in which a polynomial is written in ''nested form'': \begin &a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots + a_nx^n \\ = &a_0 + x \bigg(a_1 + x \Big(a_2 + x \big(a_3 + \cdots + x(a_ + x \, a_n) \cdots \big) \Big) \bigg). \end This allows the evaluation of a polynomial of degree with only n multiplications and n additions. This is optimal, since there are polynomials of degree that cannot be evaluated with fewer arithmetic operations. Alternatively, Horner's method and also refers to a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lill's Method
In mathematics, Lill's method is a visual method of finding the real number, real zero of a function, roots of a univariate polynomial of any degree of a polynomial, degree. It was developed by Austrian engineer Eduard Lill in 1867. A later paper by Lill dealt with the problem of complex numbers, complex roots. Lill's method involves drawing a path of straight line segments making Right angle, right angles, with lengths equal to the coefficients of the polynomial. The roots of the polynomial can then be found as the Slope, slopes of other right-angle paths, also connecting the start to the terminus, but with vertices on the lines of the first path. Description of the method To employ the method, a diagram is drawn starting at the origin. A line segment is drawn rightwards by the magnitude of the leading coefficient, so that with a negative coefficient, the segment will end left of the origin. From the end of the first segment, another segment is drawn upwards by the magnitude o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Theorem Of Algebra
The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant polynomial, constant single-variable polynomial with Complex number, complex coefficients has at least one complex Zero of a function, root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero. Equivalently (by definition), the theorem states that the field (mathematics), field of complex numbers is Algebraically closed field, algebraically closed. The theorem is also stated as follows: every non-zero, single-variable, Degree of a polynomial, degree ''n'' polynomial with complex coefficients has, counted with Multiplicity (mathematics)#Multiplicity of a root of a polynomial, multiplicity, exactly ''n'' complex roots. The equivalence of the two statements can be proven through the use of successive polynomial division. Despite its name, it is not fundamental for modern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Factorization
In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain. Polynomial factorization is one of the fundamental components of computer algebra systems. The first polynomial factorization algorithm was published by Theodor von Schubert in 1793. Leopold Kronecker rediscovered Schubert's algorithm in 1882 and extended it to multivariate polynomials and coefficients in an algebraic extension. But most of the knowledge on this topic is not older than circa 1965 and the first computer algebra systems: When the long-known finite step algorithms were first put on computers, they turned out to be highly inefficient. The fact that almost any uni- or multivariate polynomial of degree up to 100 and with coefficients of a moderate size (up to 100 bits) can be factored by modern algorithms in a few minutes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Long Division
In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalized version of the familiar arithmetic technique called long division. It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones. Sometimes using a shorthand version called synthetic division is faster, with less writing and fewer calculations. Another abbreviated method is polynomial short division (Blomqvist's method). Polynomial long division is an algorithm that implements the Euclidean division of polynomials, which starting from two polynomials ''A'' (the ''dividend'') and ''B'' (the ''divisor'') produces, if ''B'' is not zero, a '' quotient'' ''Q'' and a ''remainder'' ''R'' such that :''A'' = ''BQ'' + ''R'', and either ''R'' = 0 or the degree of ''R'' is lower than the degree of ''B''. These conditions uniquely define ''Q'' and ''R'', which means that ''Q'' and ''R'' do not depend o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computation
A computation is any type of arithmetic or non-arithmetic calculation that is well-defined. Common examples of computation are mathematical equation solving and the execution of computer algorithms. Mechanical or electronic devices (or, historically, people) that perform computations are known as ''computers''. Computer science is an academic field that involves the study of computation. Introduction The notion that mathematical statements should be 'well-defined' had been argued by mathematicians since at least the 1600s, but agreement on a suitable definition proved elusive. A candidate definition was proposed independently by several mathematicians in the 1930s. The best-known variant was formalised by the mathematician Alan Turing, who defined a well-defined statement or calculation as any statement that could be expressed in terms of the initialisation parameters of a Turing machine. Other (mathematically equivalent) definitions include Alonzo Church's '' lambda-defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder. Definition An integer n is divisible by a nonzero integer m if there exists an integer k such that n=km. This is written as : m\mid n. This may be read as that m divides n, m is a divisor of n, m is a factor of n, or n is a multiple of m. If m does not divide n, then the notation is m\not\mid n. There are two conventions, distinguished by whether m is permitted to be zero: * With the convention without an additional constraint on m, m \mid 0 for every integer m. * With the convention that m be nonzero, m \mid 0 for every nonzero integer m. General Divisors can be negative as well as positive, although often the term is restricted to posi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Synthetic Division
In algebra, synthetic division is a method for manually performing Euclidean division of polynomials, with less writing and fewer calculations than long division. It is mostly taught for division by linear monic polynomials (known as Ruffini's rule), but the method can be generalized to division by any polynomial. The advantages of synthetic division are that it allows one to calculate without writing variables, it uses few calculations, and it takes significantly less space on paper than long division. Also, the subtractions in long division are converted to additions by switching the signs at the very beginning, helping to prevent sign errors. Regular synthetic division The first example is synthetic division with only a monic linear denominator x-a. :\frac The numerator can be written as p(x) = x^3 - 12x^2 + 0x - 42 . The zero of the denominator g(x) is 3. The coefficients of p(x) are arranged as follows, with the zero of g(x) on the left: :\begin \begin \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
