1. Common factor method.

2. Use the formula method.

3. patchwork method.

Extraction of common factor method

The common factor of each term is called the common factor of each term of this polynomial. The common factor can be a monomial or a polynomial.

Specific methods: When all the coefficients are integers, the coefficients of the common factor formula should take the greatest common divisor letter of all the coefficients and the same letter of each letter, and the index of each letter should take the smallest number. When the coefficient of each term has a score, the common factor coefficient is the greatest common divisor of each score. 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 common denominator, once it is lifted, the whole family will move away, leaving 1 to look after the house, change the sign when the burden is lifted, and change the shape according to the parity. Formula method

According to the relationship between factorization and algebraic expression multiplication, we can factorize some polynomial factors by using multiplication formula. This factorization method is called formula method.

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 summation formula: a3+b3=(a+b)(a2-ab+b2)

Cubic difference formula: a3-b3=(a-b)(a2+ab+b2)

Complete cubic formula: (AB) 3 = A332B+3B2B3 = (AB) 3.

Formula: A3+B3+C3-3abc = (A+B+C) (A2+B2+C2-AB-BC-CA)

2. The basic steps of common factor method:

(1) Find the common factor

(2) Put forward 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.

Group multiplication

Grouping decomposition is a simple method to decompose factors. The following is a detailed description of this method.

There are four or more polynomials that can be decomposed into groups. There are two general forms of group decomposition: dichotomy and trichotomy.

Cross multiplication

① factorization of x2+(p+q) x+pq 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 the coefficient of 1: x2+(p+q) x+pq = (x+p) (x+q).

Example 1: x2-2x-8

=(x-4)(x+2)

② Factorization of KX2+MX+N formula.

If k=ab, n=cd and ad+bc=m, then KX2+MX+N = (AX+C) (BX+D).

Example 2: Decomposition 7x2- 19x-6

The diagram is as follows: a=7 b= 1 c=2 d=-3.

Because -3× 7 =-2 1, 1× 2 = 2, and -2 1+2=- 19,

So the original formula = (7x+2) (x-3).

Formula of cross multiplication: divide by quadratic term, divide by constant term, cross multiply and sum to get linear term.

Judgement theorem of cross multiplication: If there is a formula ax2+bx+c, if b2-4ac is a complete square number, then this formula can be decomposed by cross multiplication.

Corresponding cross multiplication and double cross multiplication can also be learned.

Removing and adding methods

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(c-a)-ab(a+b)

= BC(c-a)+ca(c-a)+BC(a+b)-ab(a+b)

=(bc+ca)(c-a)+(bc-ab)(a+b)

=c(c-a)(b+a)+b(a+b)(c-a)

=(c+b)(c-a)(a+b)。

Method of completing a square

For some polynomials that cannot be formulated, they can be fitted in a completely flat way, and then factorized by the square difference formula. This method is called matching method. It belongs to the special case of the method of splitting items and supplementing items. It should also be noted that the deformation must be carried out under the principle of equality with the original polynomial.

For example: x2+3x-40

=x2+3x+2.25-42.25

=(x+ 1.5)2-(6.5)2

=(x+8)(x-5)。

factor theorem

For the polynomial f(x), if f(a)=0, then f(x) must contain the factor x-a. 。

For example, if f(x)=x2+5x+6 and f(-2)=0, it can be determined that x+2 is a factor of x2+5x+6. (Actually, it is x2+5x+6 = (x+2) (x+3). )

Note: 1. For polynomials whose coefficients are all integers, if x = q/p(p (when p and q are coprime integers), the polynomial value is zero, then q is the divisor of constant terms and p is the divisor of the highest degree.

2. For the polynomial f (a) = 0, where b is the highest coefficient and c is a constant term, then a is the divisor of C/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. Note: don't forget to return the RMB after exchange.

For example, if you decompose (x2+x+1) (x2+x+2)-12, you can make y=x2+x, then

The original formula =(y+ 1)(y+2)- 12.

= y2+3y+2- 12 = y2+3y- 10

=(y+5)(y-2)

=(x2+x+5)(x2+x-2)

=(x2+x+5)(x+2)(x- 1)。

Synthetic division

Let the polynomial f(x)=0 and find its roots as x 1, x2, x3, ..., xn, then the polynomial can be decomposed into f (x) = a (x-x1) (x-x2) (x-x3) ... (x-xn).

For example, when decomposing 2x4+7x3-2x2- 13x+6, let 2x4 +7x3-2x2- 13x+6=0.

By comprehensive division, the roots of the equation are 0.5, -3, -2, 1.

So 2x4+7x3-2x2-13x+6 = (2x-1) (x+3) (x+2) (x-1).

Let y=f(x) be the image of function y=f(x), and find the intersection points of function image and x axis X 1, X2, X3...Xn, then the polynomial can be factorized into f (x) = f (x) = a (x-x1) (x

Compared with ⑼ method, it can avoid the complexity of solving equations, but it is not accurate enough.