What is the pending coefficient? What is the Pending Coefficient Method

The undetermined coefficient is a mathematical problem-solving method that is commonly used in high school.

One year, there was a question in the sub-question of the national college entrance examination that was like this: decomposition factor.

xx－2xy＋yy＋2x－2y－3。

Analyse. Pending coefficient method.

It is an important method in junior high school mathematics, and we use this method to solve this problem: first look at polynomials.

The quadratic term xx 2xy yy in can be decomposed into (x y)?（x－y）

Therefore, if the polynomial can be decomposed into the product of two primary factors with respect to x and y, then the two factors must be of the form (x y m) (x y n), where m and n are the coefficients to be determined, and the polynomial can be decomposed as long as the values of m and n can be found.

Untie. Let xx 2xy yy 2x 2y 3 (x y m)(x y n) xx 2xy yy (m n)x ( m n)y mn

Two polynomial identities are equal in coefficients of their corresponding terms.

Solution, get. m＝－1

n＝3xx－2xy＋yy＋2x－2y－3＝(x－y－1)(x－y＋3)

Through this example, it can be seen that the method of undetermined coefficients is used to decompose the factors by the method of undetermined coefficients, that is, the original formula is assumed to be the product of several factors according to the known conditions, and the coefficients in these factors can be represented by letters first, and their values are to be determined, because the continuous product of these factors is the same as the original formula, and then according to the principle of identity, the system of equations of the undetermined coefficients is established, and finally the value of the undetermined coefficients can be obtained by solving the system of equations.

The simplest way to solve this problem is to divide into groups and use the holistic thinking method (thinking about x y as a whole) to break down the pattern of the chop and the gall of the arrow.

Untie. Original (xx 2xy yy).

2x－2y)－3

x－y)(x－y)＋2(x－y)－3

x－y－1)(x－y＋3)

If you are a high school student, you can try to do the following question in this way:

Decomposition factor: 6xx 5xy yy x y 2

I'm sorry, I really can't think of any questions about the pending coefficients in the first year of junior high school, you can find me on hi.

A way to find unknowns. Expressing a polynomial as another new form with undetermined coefficients gives you an identity. Then, according to the properties of the identity, the equation or system of equations that the coefficients should satisfy is obtained, and then the undetermined coefficients can be found by solving the equations or systems of equations, or the relations satisfied by some coefficients can be found, and this method of solving the problem is called the undetermined coefficient method.

There is no definite coefficient yet.

The undetermined coefficient method basically does not have any technical content, it is a pure application of the general solution formula, and then the application of the derivative formula and the derivation law twice, the amount of calculation is a little larger. First, solve the homogeneous linear equation y''+4y=''The eigenequation of +4y=0 is r2+4=0, the root is 2i, and the corresponding linear gratuitous special solution is cos2x, sin2x, so y''The general solution of +4y=0 is y=c1cos2x+c2sin2x. Secondly, the nonhomogeneous linear equation y''The free term +4y=8x is 8x, which is considered 8x*e (0*x), =0.

Because =0 is not the root of the eigenequation, y''The special solution of +4y=8x is y*=(ax+b)e (0*x)=ax+b. Substituting the non-homogeneous linear equation gives 4ax+4b=8x, so 4a=8,4b=,b=0,y*=2x. So y''The general solution of +4y=8x is y=c1cos2x+c2sin2x+2x.

C1=0 and C2=1 are obtained from the initial condition. So the special solution of the original differential equation is y=sin2x+2x.

The undetermined coefficient method is a method for solving a system of linear equations, and its basic idea is to transform the original system of equations into an easy-to-solve triangular system of equations by constructing the undetermined coefficients, so as to find the solution of the system of equations.

1.Basic idea

The basic idea of the undetermined coefficient method is to transform the original system of equations into a triangular system of equations that is easy to solve by constructing the undetermined coefficients, so as to find the solution of the system of equations.

Specifically, the undetermined coefficient method represents the unknowns in the system of equations as a linear combination of some undetermined coefficients, and then substitutes these undetermined coefficients into the original system of equations to obtain a new set of equations. By eliminating and substituting a new system of equations, a triangular system of equations can be obtained, and the solution of the system of equations can be obtained.

2.Construct a coefficient to be determined

The key to the undetermined coefficient method is how to construct the undetermined coefficient. In general, the number of coefficients to be determined should be equal to the number of unknowns in the system of equations. For a general system of linear equations, the undetermined coefficients can be constructed by observing the characteristics of the coefficient matrix.

For example, if the coefficient matrix is a three-diagonal matrix, then a three-diagonal matrix of undetermined coefficients can be constructed.

3.Solve triangular equations

The undetermined coefficient method converts the original system of equations into a triangular system of equations, which can be solved by the band back method or the inverse method. The backband method solves each unknown in order from the bottom to the top, and the inverse method solves each unknown in order from the top down.

In the process of solving triangular equations, it is necessary to pay attention to whether the coefficient matrix is singular, if the coefficient matrix is singular, then the system of equations has no solution or has an infinite number of solutions.

4.Pros and cons:

The advantages of the undetermined coefficient method are that the calculation is simple, easy to understand, and suitable for general systems of linear equations. The disadvantage is that for some special systems of linear equations, it may be necessary to construct complex undetermined coefficients, and the computational complexity is larger than that of the implicit equation. In addition, the undetermined coefficient method is only applicable to systems of linear equations and cannot be used for systems of nonlinear equations.

5.Knowledge Expansion:

The undetermined coefficient method is an important method in junior high school mathematics. To decompose the factors by the method of undetermined coefficients, that is, to first assume the original formula as the product of several factors according to the known conditions, the coefficients in these factors can be expressed by letters first, and their values are to be determined, because the continuous product of these factors is the same as the original formula, and then according to the principle of identity, the system of equations of the undetermined coefficients is established.

Do you know what the pending coefficient method is.

The general steps to solve a problem using the pending coefficient method are:

1) Determine the general analytic formula for the problem with the coefficients to be determined;

2) According to the identity bar, a set of equations with undetermined coefficients are listed;

3) Solve the equation or eliminate the pending coefficients, so that the problem is solved.

For example: "Knowing that x2-5=(2-a)·x2+bx+c, find the values of a,b,c" It is not difficult to solve this problem Only Duan Qi needs to compare the coefficients of the corresponding terms in the polynomial of the right and left formulas to obtain the values of a, b, and c Here a, b, c are the coefficients to be determined, and this method of solving the problem is the coefficients to be determined

Categories: Science & Engineering.

Problem description: Same as above (verbose).

Analysis: A polynomial is expressed as another new form with a coefficient, so that an identity is obtained. Then according to the properties of the identity, the equation or system of equations that the coefficients should satisfy is obtained, and then the coefficients to be determined can be found by solving the equations or systems of equations, or the relations satisfied by some coefficients can be found.

The general steps to solve the problem using the pending coefficient method are: (1) determine the analytical formula of the problem with the undetermined coefficient; (2) According to the identity condition, a set of equations with undetermined coefficients are listed3) Solve the equation or eliminate the pending coefficients, so that the problem is solved.

For example: "Knowing that x 2-5 = (2 a a) · x 2 bx c (x 2 means the square of x), find the values of a, b, c "It is not difficult to solve this problem Just compare the coefficients of the corresponding terms in the multi-medium term of the right and left formulas, and you can get the values of a, b, and c where a, b, and c are the coefficients to be determined, and the solution to this problem is the coefficients to be determined

Steps: 1. Determine the analytical formula of the problem with the coefficients to be determined. In the example above, the analytic expression is:

2 a) · x 2 bx c 2 - according to the identity condition, list a set of equations with undetermined coefficients. In this problem, the identity condition is: 2-a=1 b=0 c=-5 Third, solve the equation or eliminate the undetermined coefficients, so that the problem is solved.

a=1 b=0 c=-5 The answer comes out.