What type of factoring to use the pending coefficients

Most of them are 3 or 4 times.

Let it be a quadratic or a quadratic multiplication.

The details are as follows:x⁴+y⁴x⁴+2x²y²+y⁴-2x²y²(x²+y²)²2x²y²

x²+y²+xy)(x²+y²-xy)

Pending coefficient method:In the process of factorization, some polynomials can be analyzed and concluded that they can be decomposed into several factors, but some coefficients in these factors have not yet been determined, and some letters can be used to represent the coefficients to be determined.

Since the polynomial is equal to the product of these factors, according to the nature of the polynomial identity, the coefficients of the corresponding terms on both sides should be equal, and the key family or several special values of the original letters in the polynomial are taken to list the equations (or systems of equations) about the undetermined coefficients, and the value of the coefficients of the undetermined letters is solved, and the method of factorization is called the undetermined coefficient method.

To decompose a factor by the method of undetermined coefficients is to set the original formula as the product of several factors according to the known conditions, and then use the polynomial identity theorem to find the value of each undetermined coefficient.

The undetermined coefficient method, a method of finding 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.

The undetermined coefficient method is an important method in junior high school mathematics. Decompose the factors with the method of undetermined coefficients, that is, 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, 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 equation system of the undetermined coefficients is established, and finally the value of the undetermined coefficients can be obtained by solving the equation system.

The simple application of the factoring theorem is actually a trick:

If the sum of the coefficients is 0, it must contain a factor (x-1); If the sum of the odd coefficients is equal to the sum of the even coefficients, then the must contain a factor (x+1) that can be introduced by a cross multiplication, since the cross multiplication is a special method of undetermined coefficients.

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

1) Determine the general analytic formula of the problem with the undetermined coefficients.

2) According to the identity condition, a set of equations with undetermined coefficients is listed.

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

Factoring: x +6x +11x+6

Let x +6x +11x+6=(x+a)(x+b)(x+c)(x+a)(x+b)(x+c).

x²+ax+bx+ab)(x+c)

x³+ax²+bx²+cx²+abx+acx+bcx+abc=x³+(a+b+c)x²+(ab+ac+bc)x+abc∴a+b+c=6

ab+ac+bc=11

abc=6 solution: a=1 b=2 c=3

x +6x +11x+6=(x+1)(x+2)(x+3) This is the undetermined coefficient method.