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:

a2-b2=(a+b)(a-b)

a2+2ab+b2=(a+b)2

a2-2ab+b2=(a-b)2

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

(2) Variance formula

1. Variance formula

Equation (1): a2-b2=(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) Factorization

1. In factorization, if there is a common factor, first raise the common factor and then decompose it further.

2. Factorization must be carried out until each polynomial factor can no longer be decomposed.

(4) Complete square formula

(1) Reversing the multiplication formula (a+b)2=a2+2ab+b2 and (a-b)2=a2-2ab+b2, we can get:

a2+2ab+b2 =(a+b)2

a2-2ab+b2 =(a-b)2

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.

Equations a2+2ab+b2 and a2-2ab+b2 are called completely flat modes.

The above two formulas are called complete square formulas.

(2) the form and characteristics of completely flat mode

① Number of projects: three projects.

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) Factorization must be decomposed until every polynomial factor can no longer be decomposed.

(5) Grouping 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 called factorization polynomial, because it does not conform to the meaning of factorization. But it is not difficult to see that these two terms have a common factor (m+n), so they can be decomposed continuously, so

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.

(6) 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. Use the formula x2 +(p+q)x+pq=(x+q)(x+p) for factorization, and pay attention to:

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

Coefficient of linear term.

2. Many people try 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 sum of two factors is exactly equal to the first-order coefficient.

3. The original polynomial is decomposed into the form of (x+q)(x+p).

(7) Multiplication and division of fractions

1. Dividing the numerator of a fraction by the common factor of the denominator is called the divisor of the fraction.

2. The purpose of score reduction is to reduce this score to the simplest score.

3. If the numerator or denominator of the fraction is a polynomial, we can first consider decomposing it into factors to get the form of factor product, and then reduce the common factor of the numerator and denominator. If the polynomial in the numerator or denominator can't decompose the factor, then we can't separate some items in the numerator and denominator.

4. Pay attention to the correct use of the sign law of power in fractional reduction, such as x-y =-(y-x), (x-y) 2 = (y-x) 2,

(x-y)3=-(y-x)3。

5. The numerator or denominator of a fraction is signed to the nth power, which can be changed into the symbol of the whole fraction according to the sign law of the fraction, and then treated as the positive even power and negative odd power of-1. Of course, the numerator and denominator of a simple fraction can be directly multiplied.

6. Pay attention to the parentheses, then the power, then the multiplication and division, and finally the addition and subtraction.

(8) Addition and subtraction of scores

1. Although general fractions and reduction are aimed at fractions, they are two opposite variants. Reduction is for one score, while general scores are for multiple scores. The approximate fraction is a simplified fraction, and the general fraction is a simplified fraction, thus unifying the denominator of the fraction.

2. Both general score and approximate score are deformed according to the basic properties of the score, and their similarity is to keep the value of the score unchanged.

3. The general denominator is written in the form of unexpanded continuous product, and the numerator multiplication is written in polynomial to prepare for further operation.

4. Total score basis: the basic nature of the score.

5. The key to general division is to determine the common denominator of several fractions.

Usually, the product of the highest power of all factors of each denominator is taken as the common denominator, which is called the simplest common denominator.

6. By analogy, get the total score of this score:

Changing several fractions with different denominators into fractions with the same mother equal to the original fraction is called the general fraction of fractions.

7. The rules for adding and subtracting fractions with the same denominator are: adding and subtracting fractions with the same denominator and adding and subtracting numerators with the same denominator.

Addition and subtraction of fractions with the same denominator, denominator unchanged, addition and subtraction of molecules, that is, the operation of fractions is transformed into the operation of algebraic expressions.

8. Fraction addition and subtraction law of different denominators: Fractions of different denominators are added and subtracted, first divided by fractions of the same denominator, and then added and subtracted.

9. Fractions with the same denominator are added and subtracted, and the denominator remains the same. Add and subtract molecules, but pay attention to each molecule as a whole, and put parentheses in due course.

10. For the addition and subtraction between the algebraic expression and the fraction, the algebraic expression is regarded as a whole, that is, it is regarded as a fraction with the denominator of 1, so as to divide.

1 1. For addition and subtraction of fractions with different denominators, first observe whether each formula is the simplest fraction. If the fraction can be simplified, it can be simplified first and then divided, which will simplify the operation.

12. As the final result, if it is a score, it should be the simplest score.

(9) One-dimensional linear equation with letter coefficient

1. One-dimensional linear equation with letter coefficient

Example: A times (a≠0) of a number is equal to B, so find this number. This number is represented by X. According to the meaning of the question, the equation ax=b(a≠0) can be obtained.

In this equation, X is unknown, and A and B are known numbers in letters. For x, the letter a is the coefficient of x and b is a constant term. This equation is a one-dimensional linear equation with letter coefficients.

The solution of the letter coefficient equation is the same as that of the numerical coefficient equation, but special attention should be paid to: multiply or divide two sides of the equation with a letter, and the value of this formula cannot be equal to zero.