Definition: transforming a polynomial into the product of several simplest algebraic expressions is called factorization of this polynomial (also called factorization).

Formula method

If the multiplication formula is reversed, some polynomials can be factorized. This method is called formula method. Square difference formula: (A+B) (A-B) = A 2-B 2, followed by A 2-B 2 = (A+B) (A-B) Complete square formula: (A+B) 2 = A 2+2AB+B 2, followed by a. 2 Note.

Factorization technique

1. Factorization factor and algebraic expression multiplication are reciprocal deformation. 2. Master factorization skills: ① The left side of the equation must be a polynomial; ② The result of factorization must be expressed in the form of product; ③ Each factor must be an algebraic expression, and the degree of each factor must be lower than that of the original polynomial; ④ Factorization factors must be decomposed until each polynomial factor can no longer be decomposed. Note: Find the common factor before decomposing the factor, and consider the coefficient and factor before determining the common factor. 3. Basic steps of common factor method: (1) Find common factor; (2) Put forward the common factor and determine another factor: ① First, determine the coefficient, and then determine the letters according to the method of determining the common factor to find the common factor; (2) The second step is to put forward the common factor and determine another factor. Pay attention to determine another factor. You can divide the original polynomial by the common factor, and the quotient is the remainder after improving the common factor. You can also use the common factor to remove each term of the original polynomial and find the remaining factors. (3) After extracting the common factor, the number of terms of another factor is the same as that of the original polynomial.

For example: AX+AY+bx+BY = A (X+Y)+B (X+Y) = (A+B) (X+Y) We divide AX and AY into a group, BX and BY into a group, and use the law of multiplication and distribution to match each other, which solves the difficulty at once. Similarly, this problem can be done. Ax+ay+bx+by = x (a+b)+y (a+b) = (a+b) (x+y) Examples: 1. 5ax+5bx+3ay+3by solution: = 5x (a+b)+3y (a). 2.x 3-x2+x- 1 solution: = (x3-x2)+(x-1) = x2 (x-1)+(x-1) = (. 3.X 2-X-Y 2-Y solution: = (X2-Y2)-(X+Y) = (X+Y) = (X+Y) (X-Y-1.

Cross multiplication

There are two situations in this method. Factorization of (1)x2+(P+Q)X+PQ formula The characteristics of this kind of quadratic trinomial are: the coefficient of quadratic term is1; Constant term is the product of two numbers; The coefficient of a linear term is the sum of two factors of a constant term. So we can directly factorize some quadratic trinomial terms with the coefficient of1:x 2+(p+q) x+pq = (x+p) (x+q). ② if k=ac and n=bd, then kx 2+MX+n = (ax+b) (CX+d). The diagram is as follows: a ╲╱ c b ╲ d For example, because1╲╱ 2-3 ╲.

Method of splitting and adding items

This method refers to disassembling one term of a polynomial or filling two (or more) terms that are opposite to each other, so that the original formula is suitable for decomposition by improving the common factor method, using the formula method or grouping decomposition method. It should be noted that the deformation must be carried out under the principle of equality with the original polynomial. For example: BC (B+C)+CA (C-A)-AB (A+B) = BC (C-A+A+B)+CA (C-A)-AB (A+B) = BC (C-A)+BC (A+B)+CA (. +(BC-ab)(a+b)= c(c-a)(b+a)+b(a+b)(c-a)=(c+b)(c-a)(a+b)。

Alternative method

Sometimes in factorization, you can choose the same part of the polynomial, replace it with another unknown, then factorize it and finally convert it back. This method is called substitution method. Correlation formula

Note: Don't forget to return RMB after the exchange. For example, when decomposing (x2+x+1) (x2+x+2)-12, you can make y = x 2+x, then the original formula = (y+ 1) (y+2)-65438.