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.
Decomposition steps:
(1) If the polynomial term has a common factor, then the common factor should be raised first;
(2) If there is no common factor, try to decompose it by formula and cross multiplication;
(3) If none of the above methods can be used for decomposition, you can try to decompose by grouping, splitting and adding items.
(4) Factorization must be carried out until every polynomial factorization can no longer be decomposed.
It can also be summarized in one sentence: "First, look at whether there is a common factor, and then look at whether there is a formula. Try cross multiplication, and grouping decomposition should be more appropriate. "
Extended data
Main methods:
1, common factor extraction method:
If every term of a polynomial has a common factor, we can put forward this common factor, so that the polynomial can be transformed into the product of two factors. This method of decomposing factors is called the improved common factor method.
Basic steps of common factor method:
(1) Find the common factor
(2) Take the common factor and determine another factor:
① The first step is to find the common factor. You can determine the coefficient first, and then determine the letters according to the method of determining the common factor.
(2) The second step is to extract the common factor and determine another factor. Note that to determine another factor, you can divide the original polynomial by the common factor, and the quotient obtained is the remaining factor after extracting the common factor, or you can use the common factor to remove each term of the original polynomial and find the remaining factor.
(3) After extracting the common factor, the number of terms of another factor is the same as that of the original polynomial.
2. Formula method:
Reverse the square difference formula and the complete square formula of the multiplication formula to obtain the factorization formula:
Square difference formula: A2-B2 = (a+b) (a-b);
Completely flat mode: A2AB+B2 = (AB) 2;
3, group decomposition method:
The method of grouping factorization is called grouping factorization, and AC+AD+BC+BD = A (C+D)+B (C+D) = (A+B) (C+D).
Its principle:
① Continuous extraction of common factors: after grouping, each group can decompose factors, and after each group decomposes factors, there are common factors among groups.
② Apply the formula method directly after grouping: After grouping, the formula can be directly applied to each group. After each group decomposes the factors, a formula is formed between the groups, and then the factors are decomposed by the formula method.
4. Cross multiplication: A2+(P+Q) A+P Q = (A+P) (A+Q).
5, equation solving method:
Factorization by solving equations, for example
X2+2x+ 1=0, and the original formula =(x+ 1)×(x+ 1) is obtained.
6, undetermined coefficient method:
Firstly, the form of factorization factor is judged, then the letter coefficient of the corresponding algebraic expression is set, and the letter coefficient is calculated, thus decomposing polynomial factor.