How to do cross multiplication in mathematics

(1) common factor method

The common factor of each term is called polynomial term.

Common factor formula

If every term of a polynomial has a common factor, it can be put forward, so that the polynomial can be transformed into the product of two factors. This method of decomposing factors is called

Methods of improving common factor

Specific methods:

When all the coefficients are integers, the coefficients of the common factor formula should take the greatest common divisor of all the coefficients; The letter takes the same letter of each item, and the index of each letter takes the smallest number; Take the same polynomial with the lowest degree.

If the first term of a polynomial is negative, a "-"sign is usually put forward to make the coefficient of the first term in brackets become positive.

When the "-"sign is put forward, the terms of the polynomial should be changed.

Formula: find the right common factor and clean it up once; The whole family moved out and left 1 to look after the house; The negative sign should be changed, and the deformation depends on parity.

For example:-am+BM+cm =-m (a-b-c);

a(x-y)+b(y-x)= a(x-y)-b(x-y)=(x-y)(a-b).

Note: Changing 2A 2+ 1/2 to 2 (A 2+ 1/4) is not a common factor.

⑵ Formula method

If the multiplication formula is reversed, some polynomials can be factorized. This method is called

Formula method

formula for the difference of square

:a^2-b^2=(a+b)(a-b)；

Perfect square trinomial

:a^2 2ab+b^2=(a b)^2；

Note: Polynomials that can be decomposed by the complete square formula must be trinomial, two of which can be written as the sum of squares of two numbers (or formulas), and the other is twice the product of these two numbers (or formulas).

Cubic sum formula

:a^3+b^3=(a+b)(a^2-ab+b^2)；

Difference of cube

:a^3-b^3=(a-b)(a^2+ab+b^2)；

Complete cubic formula

:a^3 3a^2b+3ab^2 b^3=(a b)^3.

Formula: A 3+B 3+C 3+3 ABC = (A+B+C) (A 2+B 2+C 2-AB-BC-CA)

For example: a 2

+4ab+4b^2

=(a+2b)^2。

(3) Factorization skills

1. Factorization factor and algebraic expression multiplication are reciprocal deformation.

2. 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 the common factor;

(2) Take the common factor and determine another factor:

(1) The first step to find the common factor can be determined according to the method of determining 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.

Cross multiplication

There are two situations in this method.

①x？

(

p+q

)

x+pq

Factorization of type formula

The characteristics of this kind of quadratic trinomial formula 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 decompose some quadratic trinomial factors with coefficients of 1: x?

(

p+q

)x+pq=(x+p)(x+q)

②kx？

Title used by people who don't want to specify their gender.

Factorization of n-type formula

If there is k=ac, n=bd, and there is ad+bc=m, then kx?

Title used by people who don't want to specify their gender.

n=(ax

b)(cx

d)。

The chart is as follows: