undetermined coefficient

A way to discover the unknown. The general usage is that all or part of the coefficients of a polynomial are unknown, and these coefficients are determined by using the principle that the coefficients are equal when two polynomials are the same or other known conditions, so as to get the value to be found. For example, if a known polynomial is decomposed into factors, the coefficients of some factors can be set as unknowns, and these unknowns can be obtained by using the conditions of identities. The undetermined coefficient method can also be used to solve the conic equation that has passed some points. In a broader sense, the undetermined coefficient method is a method to solve the problem by taking some constants of an analytical formula as unknowns and using known conditions to determine these unknowns. This method can be used to find the expression of function, decompose a rational fraction into the sum of several simple fractions, and find the solution of differential equation in series form.

Another commonly used mathematical method. For some mathematical problems, if the known result has a certain form, some undetermined coefficients can be introduced to represent the result, and the identity between the given formula and the result can be established through known conditions, so that the equations or equations with undetermined coefficients as elements can be obtained, and then the undetermined coefficients can be obtained. It is widely used in polynomial factorization, solving analytical expressions of functions and curve equations.

[factorization by undetermined coefficient method]

The undetermined coefficient method is an important method in junior high school mathematics. The factorization of undetermined coefficient method refers to the assumption that the original formula is the product of several factors according to known conditions. The coefficients in these factors can be expressed by letters first, and their values are undetermined. Because the continuous product of these factors is the same as the original formula, the equations of undetermined coefficients are established according to the principle of identity, and the values of undetermined coefficients can be obtained by solving the equations. Often appear in junior high school competitions.

For example, the factorization factor x -x -5x -6x-4.

Analysis: It is easy to know that this polynomial has no primary factor, so it can only be decomposed into two secondary factors.

Solution: Let X-X-5x-6x-4 = (X+AX+B) (X+CX+D) = X+(A+C) X+(AC+B+D) X+(AD+BC) X+BD.

So the solution is

Then x-x-5x-6x-4 = (x+x+1) (x-2x-4)

Polynomials are expressed by another new form of undetermined coefficients, thus an identity is obtained. Then, according to the properties of identity, the equation or equation that the coefficient should satisfy is found, and then the coefficient to be solved is found by solving the equation or equation, or the relationship that some coefficients satisfy is found. This method of solving problems is called undetermined coefficient method.

The general steps of solving problems by using the undetermined coefficient method are: (1) determining the analytical formula of the undetermined coefficient problem; (2) According to the identity condition, a group of undetermined coefficient equations are listed; (3) Solve the equation or eliminate the undetermined coefficient, thus solving the problem.

For example: "Given that X 2-5 = (2-a) X 2+BX+C (X 2 represents the square of X), we can find the values of A, B and C." It is not difficult to solve this problem. The values of a, b and c can be obtained only by comparing the coefficients of the corresponding terms in the polynomials of the right formula and the left formula.

Step: 1. Analytical formula for determining undetermined coefficients. In the above example, the analytical formula is:

(2-A) x 2+BX+C 2。 According to the identity condition, a group of equations with undetermined coefficients are listed. In this problem, the identity condition is: 2-A= 1 B=0 C=-5. Third, solve the equation or eliminate the undetermined coefficient, so that the problem is solved. A= 1 B=0 C=-5 and the answer comes out.