For example: 5x+2=7, x=1, etc.Algebra, translated into algebra, means that letters are used to replace numbers, and then popularized.With the development of mathematics, the internal meaning is extended to replace various relationships in scientific phenomena with group structures or various structures.That is to say, "algebra" is a word of "generation" in essence, which directly studies various relationships in scientific phenomena by studying various abstract structures.
junior middle schoolAlgebraIt includes four parts: number, formula, equation and functionAlgebraic expressionAnd algebraic equations are two important contents, which are both related and essentially different.If we look at their overall structure, the similarities and differences are basically similar.
Literally,Algebraic expressionIt is only different from algebraic equation in essence.Algebraic expression is a formula that connects numbers and letters representing numbers with basic operation symbols.In this way, there is an essential difference between the deformation of algebraic expression and that of algebraic equation.The transformation of algebraic expression is identical.The theoretical basis of identical deformation is the operation rules, operation propertiesbracketsBracket removal rule、FactorizationAnd other methods.The deformation of algebraic equation is the same solution deformation.
AlgebraicSymbol(signs for algebraic equations) refers to various symbols involved in the equation, including unknown number symbols and other operation symbols.
Lagrange's Algebraic Equation Solving Theory
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The history of solving unary algebraic equations can be traced back to about AD 2000Babylonian AgeIn the long history of solution, many great mathematicians have made outstanding contributions to this, including many key figures: Hualazimi, Caldano, Ferrari, Lagrange, Abel, Galois, etc;Among many mathematicians, French Lagrange is a more prominent one. He made a turning contribution to the solution of algebraic equations. Since then, the solution of algebraic equations has undergone tremendous changes, which has promoted the birth of algebra.
From 1770 to 1771, Lagrange published a 217 page paper, R é flexions sur la R é solution alg é brique des é equations.Lagrange's main contributions to solving algebraic equations are:
1) The auxiliary equation theory is proposed, that is to say, it is necessary to pre solve a quadratic auxiliary equation when solving quadratic equation, a quadratic auxiliary equation when solving cubic equation, and a cubic auxiliary equation when solving quartic equation;For solving the quadratic equation, the solution of its auxiliary equation is a function of the root of the original quadratic equation, and only one value can be taken under the replacement of its root;For solving the cubic equation, the solution of its auxiliary equation is a function of the root of the original cubic equation, and only two values can be taken under the replacement of its root;For solving the quartic equation, the solution of its auxiliary equation is a function of the root of the original quartic equation, and only three values can be taken under the replacement of its root;Therefore, the most critical step in solving the equation is to solve the auxiliary equation of the original equation. If you can pre solve a fourth auxiliary equation when solving the quintic equation, the original quintic equation may be solved;Then we can solve an equation of n degrees. If we can solve an auxiliary equation of n-1 degrees, then the original equation may be solved.
2) It is proposed to solve the algebraic equation with the idea of replacement, and the concept of resolvent is proposed, that is, solving the algebraic equation is actually to solve its auxiliary equation, so it is necessary to find a resolvent, and the number of different values taken by the resolvent under the replacement of the original equation root is the number of times of the auxiliary equation.If an appropriate resolvent can be found, then the auxiliary equation is obtained (here the coefficient of the auxiliary equation can be expressed by the coefficient of the original equation), and the solution of the original equation can be successfully obtained by solving the auxiliary equation.
Lagrange's idea of substitution for solving algebraic equations is a great turning point in the history of solving algebraic equations, which opens a new era of solving algebraic equations.He completely changed people's thinking, making mathematicians change from simply looking for algebraic skills to finding a general and universal method - the idea of replacement to solve equations;Lagrange obtained a series of important algebraic knowledge, such as the concept of domain and the rudiment of the concept of permutation group. These knowledge were properly used by later mathematicians Ruffini, Gauss, Abel, Galois, etc., which finally solved the problem of solving algebraic equations and promoted the development of algebra itself.Therefore, Lagrange's work has had a great impact on later algebraists. Until today, many mathematicians are studying Lagrange's works.[1]
