We know that algebraic multiplication and factorization are inverse deformations of each other. If the multiplication formula is reversed, the polynomial is decomposed into factors. So there are:

a^2-b^2=(a+b)(a-b)

a^2+2ab+b^2=(a+b)^2

a^2-2ab+b^2=(a-b)^2

If the multiplication formula is reversed, it can be used to factorize some polynomials. This factorization method is called formula method.

Second, the square difference formula

1, formula: A 2-B 2 = (A+B) (A-B)

2. Language: The square difference of two numbers is equal to the product of the sum of these two numbers and the difference of these two numbers. This formula is the square difference formula.

3. Complete square formula 1, multiplication formula (A+B) 2 = A 2+2AB+B 2 and (A-B) 2 = A 2-2AB+B 2 in turn.

We can get: a 2+2ab+b 2 = (a+b) 2 and a 2-2ab+b 2 = (a-b) 2. These two formulas are called complete square formulas.

That is to say, the sum of squares of two numbers, plus (or minus) twice the product of these two numbers, is equal to the square of the sum (or difference) of these two numbers.

Let the formulas A 2+2AB+B 2 and A 2-2AB+B 2 be completely flat.

2. The form and characteristics of completely flattened mode: ① Number of items: three items;

② Two terms are the sum of squares of two numbers, and the signs of these two terms are the same;

A term is twice the product of these two numbers.

3. When there is a common factor in the polynomial, the common factor should be put forward first, and then decomposed by the formula.

4. A and B in the complete square formula can represent monomials or polynomials. Here as long as the polynomial as a whole.

5. The factory must be decomposed until every polynomial factor can no longer be decomposed.

Fourthly, group decomposition method.

Let's look at the polynomial am+an+bm+bn. These four terms have no common factor, so we can't use the method of extracting common factor, and we can't use the formula method to decompose the factors.

If we divide it into two groups (am+an) and (bm+bn), these two groups can decompose the factors by extracting the common factors respectively.

Original formula =(am+an)+(bm+bn)=a(m+n)+b(m+n)

Doing this step is not factorizing polynomials, because it does not conform to the meaning of factorization. However, it is not difficult to see that these two terms have a common factor formula (m+n), so they can be decomposed continuously, so the original formula = (AM+AN)+(BM+BN) = A (M+N)+B (M+N) = (M+N) × (A+B).

This method of decomposing factors by grouping is called grouping decomposition. As can be seen from the above example, if the terms of a polynomial are grouped and their other factors are exactly the same after extracting the common factor, then the polynomial can be decomposed by group decomposition.

Five, the common factor method

1. When decomposing a polynomial by extracting the common factor, first observe the structural characteristics of the polynomial and determine the common factor of the polynomial. When the common factor of each polynomial is a polynomial, it can be converted into a monomial by setting auxiliary elements, or the polynomial factor can be directly extracted as a whole. When the common factor of the polynomial term is implicit, the polynomial should be deformed or changed in sign appropriately until the common factor of the polynomial can be determined.

2, using the formula x 2+(p+q) x+pq = (x+q) x (x+p) factorization should pay attention to:

The (1) constant term must be decomposed into the product of two factors, and the algebraic sum of these two factors is equal to the coefficient of the linear term.

(2) Many attempts have been made to decompose the constant term into the product of two factors that meet the requirements. The general steps are as follows:

(1) lists all possible situations in which a constant term is decomposed into the product of two factors;

(2) Try which two factors are equal to the first-order coefficient.

3. decompose the original polynomial into (x+q)(x+p).