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.

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: Before determining the common factor, we should consider both the coefficient and the factor. When all the coefficients are integers, the coefficients of the common factor formula should take the greatest common divisor letter of each coefficient 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 the polynomial is negative, a negative sign should be put forward to make the coefficient of the first term in brackets become positive. When the negative sign is put forward, the term of the polynomial should be changed.

Basic steps:

(1) Find the common factor;

(2) Put forward the common factor and determine another factor;

① To find the common factor, we can first determine the coefficient, and then determine the letters according to the method of determining the common factor;

(2) common factor and determining another factor. When determining another factor, you can divide the original polynomial by the common factor, and its quotient is the number left after dividing one factor by the common factor. You can also use the common factor to remove each term of the original polynomial and find the remaining other factor.

(3) After extracting the common factor, the number of terms of another factor is the same as that of the original polynomial.

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.

Second, the formula method

If the two sides of the equal sign of the multiplication formula are reversed, we can get some special polynomial factorization formulas. This factorization method is called formula method.

Third, cross multiplication.

Cross left multiplication equals the coefficient of quadratic term, right multiplication equals the constant term, and cross multiplication plus equals the linear term.

Formula: Divided into quadratic term and constant term, cross-multiplied and summed to get a linear term. (Separate the two ends and put them in the middle)

(1) The quadratic term is decomposed by cross product, and a cross product graph (with two columns) is obtained.

(2) The constant term f is decomposed into two factors and filled in the third column. It is required that the sum of the cross products of the second and third columns in the original formula is equal to ey, and the sum of the cross products of the first and third columns in the original formula is equal to DX.

(3) First, score the constant term with the first coefficient of a letter;

(4) From the first coefficient of another letter;

(5) horizontal addition and vertical multiplication.

Fourth, the rotational symmetry method

When the topic is rotational symmetry type, it can be decomposed by rotational symmetry method.

Five, group decomposition method

Factories that cannot be directly decomposed by common factor method and formula decomposition method are decomposed by group decomposition method, which is called group decomposition method. There are four or more polynomials that can be decomposed into groups. There are two general forms of group decomposition: dichotomy and trichotomy.

Sixth, the method of deleting and adding items.

Two terms (or several terms) that decompose or fill a polynomial are opposite to each other, so that the original formula is suitable for common factor method, formula method or group decomposition method. This method of factorization is called factorization. It should be noted that the deformation must be carried out under the principle of equality with the original polynomial.

Seven, matching method

For some polynomials that cannot be formulated, we can match them in a completely flat way, and then decompose their factors with the square difference formula. This factorization 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.