Nov 14, · blogger.com has free homework help, math tutors, problem solvers and lessons. It is broken down into four sections: Pre-Algebra, Algebra I, Algebra II, and Geometry Each section features calculators, lessons, and a place to submit questions for free math help Welcome to College Pre-Algebra help from blogger.com Get the exact online tutoring and homework help you need. We offer highly targeted instruction and practice covering all lessons in College Pre-Algebra. Start now for free! Welcome to Pre-Algebra help from blogger.com Get the exact online tutoring and homework help you need. We offer highly targeted instruction and practice covering all lessons in Pre-Algebra. 31 Prime Factorization Prime Factorization, Prime Factor, what is the prime factorization, what is prime factorization, how to do prime factorization
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In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; [3] it is a unifying thread of almost all of mathematics. The more basic parts of algebra are called elementary algebra ; the more abstract parts are called abstract algebra or modern algebra.
Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values.
Algebra gives methods for writing formulas and solving equations that are much clearer and easier than the older method of writing everything out in words. The word algebra is also used in certain specialized ways.
A special kind of mathematical object in abstract algebra is called an "algebra", and the word is used, for example, in the phrases linear algebra and algebraic topology.
In his work, the term al-jabr referred to the operation of moving a term from one side of an equation to the other, المقابلة al-muqābala "balancing" referred to adding equal terms to both sides. Shortened to just algeber or algebra in Latin, the word eventually entered the English language during the 15th century, from either Spanish, Italian, or Medieval Latin.
It originally referred to the surgical procedure of setting broken or dislocated bones. The mathematical meaning was first recorded in English in the 16th century. The word "algebra" has several related meanings in mathematics, as a single word or with qualifiers. Algebra began with computations similar to those of arithmeticwith letters standing for numbers.
For example, in the algebra homework help factorization equation. That is to say, to find all the solutions of the equation. Historically, and in current teaching, the study of algebra starts with the solving of equations, such as the quadratic equation above.
Then more general questions, such as "does an equation have a solution? These questions led extending algebra to non-numerical objects, such as permutationsvectorsmatricesand polynomials. The structural properties of these non-numerical objects were then abstracted into algebraic structures such as groupsringsalgebra homework help factorization, and fields.
Before the 16th century, mathematics was algebra homework help factorization into only two subfields, arithmetic and geometry. Even though some methods, which had been developed much earlier, may be considered nowadays as algebra, the emergence of algebra and, soon thereafter, of infinitesimal calculus as subfields of mathematics only dates from the 16th or 17th century. From the second half of the 19th century on, many new fields of mathematics appeared, most of which made use of both arithmetic and geometry, and almost all of which used algebra, algebra homework help factorization.
Today, algebra has grown until it includes many branches of mathematics, as can be seen in the Mathematics Subject Classification [8] where none of the first level areas two digit entries is called algebra. Today algebra includes section General algebraic systems, Field theory and polynomialsCommutative algebraLinear and multilinear algebra ; matrix theoryAssociative rings and algebrasNonassociative rings and algebrasCategory theory ; homological algebraK-theory and Group theory.
Algebra is also used extensively in Number theory and Algebraic geometry, algebra homework help factorization. The roots of algebra algebra homework help factorization be traced to the ancient Babylonians[9] who developed an advanced arithmetical system with which they were able to do calculations in an algorithmic fashion.
The Babylonians developed formulas to calculate solutions for problems typically solved today by using linear equationsquadratic equationsand indeterminate linear equations. By contrast, most Egyptians of this era, as well as Greek and Chinese mathematics in the 1st millennium BC, usually solved such equations by geometric methods, such as those described in the Rhind Mathematical Papyrusalgebra homework help factorization, Euclid's Elementsand The Nine Chapters on the Mathematical Art.
The geometric work of the Greeks, typified in the Elementsprovided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations, algebra homework help factorization, although this would not be realized until mathematics developed in medieval Islam. By the time of PlatoGreek mathematics had undergone a drastic change.
The Greeks created a geometric algebra where terms were represented by sides of geometric objects, algebra homework help factorization, usually lines, that had letters associated with them. These texts deal with solving algebraic equations[11] and have led, in number theoryto the modern notion of Diophantine equation.
Earlier traditions discussed above had a direct influence on the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī c. He later wrote The Compendious Book on Calculation by Completion and Balancingwhich established algebra as a mathematical discipline that is independent of geometry and arithmetic. The Hellenistic mathematicians Hero of Alexandria and Diophantus [13] as well as Indian mathematicians such as Brahmaguptacontinued the traditions of Egypt and Babylon, though Diophantus' Arithmetica and Brahmagupta's Brāhmasphuṭasiddhānta are on a higher level.
Although Diophantus and the Babylonians used mostly special ad hoc methods to solve equations, Al-Khwarizmi's contribution was fundamental, algebra homework help factorization.
He solved linear and quadratic equations without algebraic symbolism, negative numbers or zerothus he had to distinguish several types of equations. In the context where algebra is identified with the theory of equationsthe Greek mathematician Diophantus algebra homework help factorization traditionally been known as the "father of algebra" and in the context where it is identified with rules for manipulating and solving equations, Persian mathematician al-Khwarizmi is regarded as "the father of algebra".
Those who support Diophantus point to the fact that the algebra found in Al-Jabr is slightly more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while Al-Jabr is fully rhetorical. He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems".
Another Persian mathematician Omar Khayyam is credited with identifying the foundations of algebraic geometry and found the general geometric solution of the cubic equation. His book Treatise on Demonstrations of Problems of Algebrawhich laid down the principles of algebra, is part of the body of Persian mathematics that was eventually transmitted to Europe, algebra homework help factorization. In the 13th century, the solution of a cubic equation by Fibonacci is representative of the beginning of a revival in European algebra.
Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī — took "the first steps toward the introduction of algebraic symbolism". He also computed Σ n 2Σ n 3 and used the method of successive approximation to determine square roots. François Viète 's work on new algebra at the close of the 16th century was an important step towards modern algebra. InRené Descartes published La Géométrieinventing analytic geometry and introducing modern algebraic notation, algebra homework help factorization.
Another key event in the further development of algebra was the general algebraic solution of the cubic and quartic equations, developed in the midth century. The idea of a determinant was developed by Japanese mathematician Seki Kōwa in the 17th century, followed independently by Gottfried Leibniz ten years later, for the purpose of solving systems of simultaneous linear equations using matrices. Gabriel Cramer also did some work on matrices and determinants in the 18th century.
Permutations algebra homework help factorization studied by Joseph-Louis Lagrange in his paper " Réflexions sur la résolution algébrique des équations " devoted to solutions of algebraic equations, in which he introduced Lagrange resolvents. Paolo Ruffini was the first person to develop the theory of permutation groupsand like his predecessors, also in the context of solving algebraic equations.
Abstract algebra was developed in the 19th century, deriving from the interest in solving equations, initially focusing on what is now called Galois theoryand on constructibility issues. Augustus De Morgan discovered relation algebra in his Syllabus of a Proposed System of Logic.
Josiah Willard Gibbs developed an algebra of vectors in three-dimensional space, and Arthur Cayley developed an algebra of matrices this is a noncommutative algebra. Some subareas of algebra have the word algebra in their name; linear algebra is one example. Others do not: group theoryring theoryand field theory are examples. In this section, algebra homework help factorization, we list some areas of mathematics with the word "algebra" in the name, algebra homework help factorization.
Elementary algebra is the most basic form of algebra. It is taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic.
In algebra, numbers are algebra homework help factorization represented by symbols called variables such as anxy or z.
This is useful because:. A polynomial is an expression that is the sum of a finite number of non-zero termsalgebra homework help factorization, each term consisting of the product of a constant and a finite number of variables raised to whole number powers.
A polynomial expression is an expression algebra homework help factorization may be rewritten as a polynomial, by using commutativity, associativity and distributivity of addition and multiplication. A polynomial function is a function that is defined by a polynomial, or, equivalently, by a polynomial expression. The two preceding examples define the same polynomial function.
Two important and related problems in algebra are the factorization of polynomialsthat is, expressing a given polynomial as a product of other polynomials that cannot be factored any further, and the computation of polynomial greatest common divisors.
A related class of problems is finding algebraic expressions for the roots of a polynomial in a single variable, algebra homework help factorization.
Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts.
Here are the listed fundamental concepts in abstract algebra. Sets : Rather than just considering the different types of numbersabstract algebra deals with the more general concept of sets : a collection of all objects called elements selected by property specific for the set. All collections of the familiar types of numbers are sets. Set theory is a branch of logic and not technically a branch of algebra.
The notion of binary operation is meaningless without the set on which the operation is defined. Identity elements : The numbers zero and one are abstracted to give the notion of an identity element for an operation. Zero is the identity element for addition and one is the identity element for multiplication. Not all sets and operator combinations have an identity element; for example, the set of positive natural numbers 1, 2, 3, has no identity element for addition, algebra homework help factorization.
Inverse elements : The negative numbers give rise to the concept of inverse elements. Associativity : Addition of integers has a property called associativity. That is, the grouping of the numbers to be added does not affect the sum. This property is shared by most binary operations, but not subtraction or division or octonion multiplication. Commutativity : Addition and multiplication of real numbers are both commutative. That is, the order of the numbers does not affect the result.
This property does not hold for all binary operations. For example, matrix multiplication and quaternion multiplication are both non-commutative. Combining the above concepts gives one of the most important structures in mathematics: a group. For example, the set of integers under the operation of addition is a group. The non-zero rational numbers form a group under multiplication. The integers under the multiplication operation, however, do not form a group. This is because, in general, the multiplicative inverse of an integer is not an integer.
The theory of groups is studied in group theory. A major result in algebra homework help factorization theory is the classification of finite simple groupsmostly published between about andalgebra homework help factorization, which separates the finite simple groups into roughly 30 basic types. Algebra homework help factorizationquasi-groupsand monoids structure similar to groups, but more general.
They comprise a set and a closed binary operation but do not necessarily satisfy the other conditions. A semi-group has an associative binary operation but might not have an identity element. A monoid is a semi-group which does have an identity but might not have an inverse for every element.
A quasi-group satisfies a requirement that any element can be turned algebra homework help factorization any other by either a unique left-multiplication or right-multiplication; however, the binary operation might not be associative.
Factor Polynomials - Understand In 10 min
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Glencoe Algebra 1 Answers ISBN: This is a comprehensive textbook that can help the student better understand the entire algebra topic. This textbook can help you understand each and every topic in algebra in a very comprehensive manner. We will help you with an overview of each and every chapter given in Glencoe algebra 1 algebra homework help rational equations rate easy way to find LCM iowa algebra aptitude test scoring guide decimals for beginners complex rational expression real life examples Factorization grade nine beginning algebra ellipse changing into form for graphing mcdougal littell algebra 2 book answers Nov 14, · blogger.com has free homework help, math tutors, problem solvers and lessons. It is broken down into four sections: Pre-Algebra, Algebra I, Algebra II, and Geometry Each section features calculators, lessons, and a place to submit questions for free math help
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