Journal Details
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Journal
eISSN
2444-8656
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01 Jan 2016
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2 times per year
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Open Access

# Algebraic Equations in Educational Model of College Physical Education Course Education

###### Accepted: 14 May 2022
Journal Details
Format
Journal
eISSN
2444-8656
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Introduction

Sports integrate politics, economy, education, technology, and management. It has become an inseparable part of society and the individual. Sports represent the development level of a country and a nation. Especially today's competitive sports is a fierce sports competition and a technological battle. The development of sports research has penetrated mathematics [1]. As the basis of natural science and social science, mathematics is also infiltrating into various areas of sports science research at an increasingly rapid speed. The mathematical application of Chinese sports science has gradually stepped out of the “primary stage” dominated by on-site statistics and fuzzy mathematical methods. In turn, the field has grown to the point where it is based on principles such as kinetics and physiology. Scholars began to use modern mathematical methods such as calculus equations, operations research, numerical analysis, and modern scientific methods and synergy to explore the inner relationship of sports. We can reveal the essential characteristics and changing laws of sports by establishing and studying mathematical models of objective objects.

The history of solving algebraic equations in one variable can be traced back to the Babylonian era around 2000 AD. Many great mathematicians have made outstanding contributions to this [2]. Among them are many important figures: Khwarizmi, Cardano, Ferrari, Lagrange, Abel, Galois, etc. Among the many mathematicians, Lagrange of France is a more prominent one. He made a transformative contribution to the solution of algebraic equations. The resolution of algebraic equations has changed dramatically since then. And thus promote the new life of algebra.

Lagrangian mathematical model of algebraic equations

For the general equation, we can solve it according to the following procedure:

Solve Equation xn + alxn−1 + alxn−2 + … + an−lx+a|n = 0

The first step is to select a suitable pre-solution formula u.

The second step y = φ(u). (τ) y can only take n-1 different values of y1, y2, ⋯, yn−1 when τSn. From this, we can determine the auxiliary equation yn−1 + A1 yn−2 + A2yn−3 +⋯+ An−2y + An−1 = 0 of the original equation [5]. Where Ai is a rational function of the coefficients of the original equation.

The third step solves y1, y2,⋯, yn−1 and then obtains u1, u2,⋯, un−1.

Step 4 Simultaneous Equations ${u1=u=c1x1+c2x2+⋯+cnxnu2=τ2(u)=c1xτ2(1)+c2xτ2(2)+⋯+cnxτ2(n)⋯un−1=τn−1(u)=c1xτ2(t)+c2xτn−1(2)+⋯+cnxτn−1(n)−a1=x1+x2+x3+⋯+xn$ \left\{\matrix{{u_1} = u = {c_1}{x_1} + {c_2}{x_2} + \cdots + {c_n}{x_n} \hfill \cr {u_2} = {\tau _2}(u) = {c_1}{x_{{\tau _2}\left(1 \right)}} + {c_2}{x_{{\tau _2}\left(2 \right)}} + \cdots + {c_n}{x_{{\tau _2}\left(n \right)}} \hfill \cr \cdots \hfill \cr {u_{n - 1}} = {\tau _{n - 1}}(u) = {c_1}{x_{{\tau _2}\left(t \right)}} + {c_2}{x_{{\tau _{n - 1}}(2)}} + \cdots + {c_n}{x_{{\tau _{n - 1}}(n)}} - {a_1} = {x_1} + {x_2} + {x_3} + \cdots + {x_n} \hfill \cr} \right.

Solving this system of equations yields the root x1, x2, x3,⋯, xn of the original equation.

The above program has no problem-solving low-order (second, third, and fourth) equations, but it is not easy to find u and y = φ(u) for higher-order (greater than fifth) equations. And the number of auxiliary equations about y sometimes encountered in solving will also be higher [6]. To this end, Lagrangian gives a general method for dealing with higher-order equations - this is the descending order. If the degree of the auxiliary equation is large, some additional equations need to be added to divide r into smaller ones. simply expressed as ${σ∈H1|σ(u)=u}=I(u)=H↔presolution formula u∩r3{σ∈H2|σ(y)=u}=I1(y)=H1↔Auxillary equation for y∩r3{σ∈Sn|σ(Φ)=Φ}=I2(φ)=H2↔Auxillary equation for φ∩r1Sn$ \eqalign{& \left\{{\sigma \in {H_1}|\sigma (u) = u} \right\} = I(u) = H \leftrightarrow {\rm{presolution}}\,{\rm{formula}}\,u \cr & \cap {r_3} \cr & \left\{{\sigma \in {H_2}|\sigma (y) = u} \right\} = {I_1}(y) = {H_1} \leftrightarrow {\rm{Auxillary}}\,{\rm{equation}}\,{\rm{for}}\,y \cr & \cap {r_3} \cr & \left\{{\sigma \in {S_n}|\sigma (\Phi) = \Phi} \right\} = {I_2}(\varphi) = {H_2} \leftrightarrow {\rm{Auxillary}}\,{\rm{equation}}\,{\rm{for}}\,\varphi \cr & \cap {r_1} \cr & {S_n} \cr}

Where r1, r2, r3 is less than the degree n of the equation. So we get more auxiliary equations. If all these additional equations are solvable, the original equation must be solvable [7]. This process can be simply expressed as $Presolution u↔H=I(u)={τ(u)=u|τ∈H1}Auxillary Equation↔H1=I1(y)={τ(y)=y|τ∈H2} for y∩Auxillary Equation↔H2=I2(φ)={τ(φ)=φ|τ∈H3} for φ∩Auxillary Equation↔H3=I3(μ)={τ(μ)=μ|τ∈H4} for μ∩⋯∩Sn$ \eqalign{& {\rm{Presolution}}\,u \leftrightarrow H = I(u) = \left\{{\tau (u) = u|\tau \in {H_1}} \right\} \cr & {\rm{Auxillary}}\,{\rm{Equation}} \leftrightarrow {H_1} = {I_1}(y) = \left\{{\tau (y) = y|\tau \in {H_2}} \right\}\,{\rm{for}}\,y \cr & \cap \cr & {\rm{Auxillary}}\,{\rm{Equation}} \leftrightarrow {H_2} = {I_2}(\varphi) = \left\{{\tau (\varphi) = \varphi |\tau \in {H_3}} \right\}\,{\rm{for}}\,\varphi \cr & \cap \cr & {\rm{Auxillary}}\,{\rm{Equation}} \leftrightarrow {H_3} = {I_3}(\mu) = \left\{{\tau (\mu) = \mu |\tau \in {H_4}} \right\}\,{\rm{for}}\,\mu \cr & \cap \cr & \cdots \cr & \cap \cr & {S_n} \cr}

If the above conclusions are accurate, we will get a subgroup sequence: HH1H2H3 ⊂ ⋯ ⊂ Sn. Thus Lagrange completed the idea of solving equations of arbitrary degree.

Lagrange's use of the idea of permutation to solve algebraic equations is a significant turning point in the history of algebraic equation solving. It opens up a new era of algebraic equation solving. He revolutionized people's thinking, turning mathematicians from simply looking for algebraic tricks to solve equations to looking for a general method. Lagrange uses the idea of permutation to solve equations [8]. Lagrange derived a series of crucial algebraic knowledge. For example, the understanding of the concept of field and the prototype of the permutation group was used correctly by later mathematicians such as Ruffini, Gauss, Abel, Galois, etc. In this way, the problem of solving algebraic equations is finally solved and promotes the development of algebra itself. So Lagrange's work had a considerable impact on later algebraists.

The influence of Lagrange's algebraic equation solving theory
The influence of Lagrange's algebraic equation solving theory on Ruffini

Ruffini proved that the general equation does not have the so-called Lagrangian resolution to satisfy an auxiliary equation of degree lower than fifth. This means that the use of Lagrangian pre-solvers does not reduce the degree of higher-order equations. Ruffini examines the rational functions of the roots of general quadratic equations [9]. If p is the number of the process that remains unchanged under the replacement of the original equation roots, of course, p must be a factor of n!. Ruffini delved into the position of the p numbers that make the rational function invariant under the permutation of the original equation roots. He pointed out that the quintic equation $n!p$ {{n!} \over p} could only be 2 or 5 or 6, not 3 or 4. There will be no 3rd or 4th order auxiliary equations that satisfy the Lagrangian resolution formula. If $n!p$ {{n!} \over p} is 2, it must be divisible by 5. This is not possible. If $n!p$ {{n!} \over p} is 5, a quintic equation that satisfies the resolved procedure exists. But we can't reduce it to a two-term equation like B anyway, so the problem remains unsolvable.

The influence of Lagrange's algebraic equation solving theory on Gaussian

Lagrange points the way to the solution of algebraic equations. He thought that the idea of permutation should be followed, but Ruffini declared that there is no radical solution to the general quintic equation. But that doesn't mean Lagrange's theory is wrong. For example, the bisector equation of the form Xn −1 = 0. He solved the bisect equation of a degree less than 11. Gauss became a blockbuster by solving the 17th-degree equation, so how did Gauss deal with it? The Gaussian approach divides the prime-order cyclotomic equation into lower-order equations [10]. Gauss realized Lagrangian's ideal on the equation of the circle. He solves the equation of a process by successfully determining the function of the roots (pre-solution).

Gaussian studies the problem from the root of the modulo p. Any prime number p, ∃gZ such that, for any k ≡ 0(mod p),, ∃iZ, such an integer g is called a fundamental root modulo p. Let x1, x2, x3, x4, x5, x6, x7 be the root of the septate equation| x7 −1 = 0. We can quickly get x1=ζ31−1 =ζ, x2=ζ32−1 =ζ3, x3=ζ33−1 =ζ9 =ζ2, x4=ζ34−1 =ζ27 = ζ6, x5=ζ35−1 =ζ81 = ζ4, x6 = ζ36−1 =ζ243 = ζ5, x7=ζ37−1 =ζ727 = ζ by taking g = 3, xi = ζgi−1.

The group G = {1,τ, τ2,τ3,τ4,τ5} can be found, and the group x7 −1 = (x −1)(x6 + x5 + x4 + x3 + x2 + x +1), so solving equation x7 −1 = 0 is solving the equation x6 + x5 + x4 + x3 + x2 + x +1 = 0. The equation is hexadecimal. Lagrangians can reduce the degree for equations of degree 6=2×3 or 6=3×2. Therefore, taking the presolution u = x1, it can only take two values under all permutations of: $I(u=x1)={1}∩ r1=3I(β=x1+x3+x5)=A={1,τ2,τ4}∩ r=2G={1,τ,τ2,τ3,τ4,τ5}$ \eqalign{& I\left({u = {x_1}} \right) = \left\{1 \right\} \cr & \cap \,\,\,{r_1} = 3 \cr & I\left({\beta = {x_1} + {x_3} + {x_5}} \right) = A = \left\{{1,{\tau ^2},{\tau ^4}} \right\} \cr & \cap \,\,\,\,\,\,\,r = 2 \cr & G = \left\{{1,\tau,{\tau ^2},{\tau ^3},{\tau ^4},{\tau ^5}} \right\} \cr}

Where β= x1 + x3 + x5 under all transformations of G, we can only take two values: $1 β=β1=x1+x3+x5τ β=β2=x2+x4+x6$ \eqalign{& 1\,\,\,\,\,\,\beta = {\beta _1} = {x_1} + {x_3} + {x_5} \cr & \tau \,\,\,\,\,\beta = {\beta _2} = {x_2} + {x_4} + {x_6} \cr}

Here we need to set its root as β1, β2 to get the quadratic auxiliary equation about β, then $β1+β2=x1+x2+x3+x4+x5+x6=−1β1×β2=(x1+x3+x5)(x2+x4+x6)=(ζ+ζ2+ζ4)(ζ3+ζ6+ζ5)=ζ4+1+ζ6+ζ5+1+1+ζ3+ζ2)=3−1=2$ \eqalign{& {\beta _1} + {\beta _2} = {x_1} + {x_2} + {x_3} + {x_4} + {x_5} + {x_6} = - 1 \cr & {\beta _1} \times {\beta _2} = \left({{x_1} + {x_3} + {x_5}} \right)\left({{x_2} + {x_4} + {x_6}} \right) = \cr & \left({\zeta + {\zeta ^2} + {\zeta ^4}} \right)\left({{\zeta ^3} + {\zeta ^6} + {\zeta ^5}} \right) = \cr & {\zeta ^4} + 1 + {\zeta ^6} + {\zeta ^5} + 1 + 1 + {\zeta ^3} + {\zeta ^2}) = 3 - 1 = 2 \cr}

From this we can obtain the quadratic auxiliary equation for β as y2 + y + 2 = 0. We can quickly solve its root β1, β2.

Where u = x1 can only take three values under all permutations of A $1 u=u1=x1τ2 u=u2=x3τ4 u=u3=x5$ \eqalign{& 1\,\,u = {u_1} = {x_1} \cr & {\tau ^2}\,\,u = {u_2} = {x_3} \cr & {\tau ^4}\,\,u = {u_3} = {x_5} \cr}

Here we need to set its root as u1, u2, u3 to get the cubic auxiliary equation about u, then $u1+u2+u3=x1+x3+x5=β1u1u2+u1u3+u2u3=x1x3+x3x5+x1x5=ζ3+ζ5+ζ6=x2+x4+x6=β2u1u2u3=x1x3x5=ζζ2ζ4=1$ \eqalign{& {u_1} + {u_2} + {u_3} = {x_1} + {x_3} + {x_5} = {\beta _1} \cr & {u_1}{u_2} + {u_1}{u_3} + {u_2}{u_3} = {x_1}{x_3} + {x_3}{x_5} + {x_1}{x_5} = {\zeta ^3} + {\zeta ^5} + {\zeta ^6} = {x_2} + {x_4} + {x_6} = {\beta _2} \cr & {u_1}{u_2}{u_3} = {x_1}{x_3}{x_5} = \zeta {\zeta ^2}{\zeta ^4} = 1 \cr}

At this point we get the cubic auxiliary equation u3β1u2 + β2u −1 = 0 about u, and its root u is easy to get. At this point we also get the origins of the original equation.

Of course, we can also use the second option. It's just that you need to solve a cubic equation first and then a quadratic equation. Its method is $I(u=x1)={1}∩ r1=2I(β=x1+x4)=A={1,τ3}∩ r=3G={1,τ,τ2,τ3,τ4,τ5}$ \eqalign{& I\left({u = {x_1}} \right) = \left\{1 \right\} \cr & \cap \,\,\,\,{r_1} = 2 \cr & I\left({\beta = {x_1} + {x_4}} \right) = A = \left\{{1,{\tau ^3}} \right\} \cr & \cap \,\,r = 3 \cr & G = \left\{{1,\tau,{\tau ^2},{\tau ^3},{\tau ^4},{\tau ^5}} \right\} \cr}

Finally, the roots of the original secret equation can also be found.

The solution process of Gauss's famous 17th-degree circle equation is roughly similar to the above. We can simply express it as $I(u=x1)={1}∩ r1=2I(α=x1+x9)=A={1,τ8}∩ r2=2I(β=x1+x5+x9+x13)=B={1,τ4,τ8,τ12}∩ r3=2I(γ=x1+x3+x5+x7+x9+x11+x13+x15)=C={1,τ2,τ4,τ6,τ8,τ10,τ12,τ14}∩ r4=2G={1,τ,τ2,τ3,τ4,⋯,τ14,τ15}$ \eqalign{& I\left({u = {x_1}} \right) = \left\{1 \right\} \cr & \cap \,\,\,\,{r_1} = 2 \cr & I\left({\alpha = {x_1} + {x_9}} \right) = A = \left\{{1,{\tau ^8}} \right\} \cr & \cap \,\,{r_2} = 2 \cr & I\left({\beta = {x_1} + {x_5} + {x_9} + {x_{13}}} \right) = B = \left\{{1,{\tau ^4},{\tau ^8},{\tau ^{12}}} \right\} \cr & \cap \,\,{r_3} = 2 \cr & I\left({\gamma = {x_1} + {x_3} + {x_5} + {x_7} + {x_9} + {x_{11}} + {x_{13}} + {x_{15}}} \right) = C = \cr & \left\{{1,{\tau ^2},{\tau ^4},{\tau ^6},{\tau ^8},{\tau ^{10}},{\tau ^{12}},{\tau ^{14}}} \right\} \cr & \cap \,\,{r_4} = 2 \cr & G = \left\{{1,\tau,{\tau ^2},{\tau ^3},{\tau ^4}, \cdots,{\tau ^{14}},{\tau ^{15}}} \right\} \cr}

Here g still takes 3. We decompose the equation x16 + x15 +⋯+ x3 + x2 + x1 + x +1 = 0 of degree 16 into 2×2×2×2. The process of solving the equation is roughly similar to the above. So in the end, Gauss got the roots of the 17th-degree bisector equation. It is easy to see that Gauss's method of solving the circle equation is Lagrangian's method. The general cycloid equation is expressed as xn −1 = 0 (n is a prime number). Assuming n −1 = 2× r1 × r2 ×⋯× rm we can obtain its roots by the above method, Gauss declares that any degree of the circular equation is solvable.

Gauss is very clear about the connotation of the theory of Lagrangian algebraic equations, including that the pre-solved formula should be selected and the degree of the equation should be reduced. His procedure for solving the circle equation is an implementation of the Lagrangian method. Lagrange's theory of solving algebraic equations is validated here by Gauss. Therefore, without Lagrange's algebraic equation solving theoretical Gaussian, it is impossible to obtain the success of solving the 17th-degree centroid equation.

The influence of Lagrange's algebraic equation solving theory on Galois

The most significant victory of the Lagrangian algebraic equation solving theory directly led to the emergence of the Galois theory. Its influence on Galois was enormous. The first is spiritual motivation [11]. Lagrange's solving algebraic equations stimulated Galois's idea to the greatest extent. Lagrange's study of the solution of equations of the fifth degree and above led Galois to be the first to realize that the different representations of numerical values must be taken into account when studying particular equations. So permutation doesn't change the value of numbers, only their indication.

Second, many algebraists say that Galois changed the face of solving algebraic equations. This has led to pure algebra in mathematics since him. Lagrange divided algebra into two parts: mathematics and pure algebra. Galois continued Lagrange's work. He further developed algebra based on Lagrange.

Finally, Lagrange gave Galois great help in solving the algebraic equations. Some of the fundamental theories of Lagrange still appear in Galois's view. Some of Galois's ideas have emerged in Lagrange's work. Such as presolution. Lagrange requires three conditions when discussing the pre-solution formula selection: 1) The pre-solution procedure should be a rational expression of the original equation roots and known numbers. 2) Related numbers and known numbers can rationally express each root of the equation. 3) must be the root of a solvable equation. 1) is the criterion for describing the choice of presolution formula. 2) and 3) describe the properties that the pre-solution should satisfy. It is undeniable that Lagrange did select the pre-solution recipe and solve the algebraic equation according to the above three points. This may be one of the reasons why he didn't get the final victory. Galois realized that choosing the pre-solved formula is enough to satisfy 1) and 2) for an equation. And there is no pre-solution formula that needs to satisfy the three conditions simultaneously. Therefore, Galois referred to Lagrange's research theory on the problem of pre-solution selection.

In addition, it was said in the previous article that Lagrange pointed out the direction for solving algebraic equations, but he did not point out specific measures. That is to say, according to Lagrange's idea, it should be possible to find the root of the equation, and Galois did it. Under the guidance of Lagrange, he found this specific path to the end. According to the idea of solving Lagrange's theoretical algebraic equations, a subgroup sequence is finally obtained [12]. The core of Galois's theory is to find such a subgroup sequence. In any respect, Lagrangian has the most significant influence on Galois. Lagrange established the framework of the idea, and it is within this framework Galois won the victory of solving algebraic equations.

Conclusion

The sports mathematical model originates from the original but is higher than the sports prototype. It is an abstraction and simplification of a discernible sports archetype. The purpose of establishing a sports mathematical model is to carry out sports scientific research better and then solve practical problems. So we have to complete the calculation of the mathematical model. However, these tasks are usually incompetent with humans and must be done with the help of computers. The application of computers makes the research of sports mathematical models have a particular foundation. But the direct “addition” of mathematical models and computers does not always meet practical needs. The size of the system, the amount of data, and the time constraints require us to have suitable computing methods. It is this that drives the development of mathematics and sports science. It promotes the modernization of sports science. Mathematical models set up a new bridge for further integrating disciplines such as sports science and computer science.

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