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.

(2) Variance formula

1. Variance formula

Equation (1): 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) 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) By reversing the multiplication formula (a+b) 2 = a 2+2ab+b 2 and (a-b) 2 = a 2-2ab+b 2, we can get:

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

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

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

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 until the common factor of the polynomial can be determined.

2. factorization with the formula X 2+(P+Q) X+PQ = (X+Q) (X+P), it should be noted that:

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.

(viii) 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.

Mathematics (1) in the second day of junior high school should know the knowledge points that should be known.

factoring

1. Factorization: decomposing a polynomial into the product of several algebraic expressions is called decomposing this polynomial; Note: Factorization and multiplication are two opposite transformations.

2. Factorial decomposition method: common methods include extracting common factors, formula method, grouping decomposition method and cross multiplication.

3. Determination of common factor: What is the greatest common factor of the coefficient? The lowest power of the same factor.

Pay attention to the formula: a+b = b+a; a-b =-(b-a)； (a-b)2 =(b-a)2； (a-b)3=-(b-a)3。

4. The formula of factorization:

(1) square difference formula: A2-B2 = (a+b) (a-b);

(2) Complete square formula: A2+2ab+B2 = (a+b) 2, A2-2ab+B2 = (a-b) 2.

5. Matters needing attention in factorization:

(1) The general order of selecting factorization methods is: one extraction, two formulas, three grouping and four crossover;

(2) When using factorization formula, special attention should be paid to the integrity of letters in the formula;

(3) The final result of factorization needs factorization until every factorization cannot be decomposed;

(4) The final result of factorization requires the first sign of each factor to be positive;

(5) The final result of factorization needs to be sorted;

(6) The final result of factorization requires that the same factor be written as a power.

6. Problem solving skills of factorization: (1) transposition arrangement, bracket arrangement or bracket arrangement; (2) negative sign; (3) Total symbol change; (4) exchange RMB; (5) formula; (6) treating the same formula as a whole; (7) flexible grouping; (8) extracting the fractional coefficient; (9) expand some or all of the brackets; (10) Dismantle or supplement.

7. Completely flat mode: A polynomial that can be reduced to (m+n)2 is called completely flat mode; For quadratic trinomial x2+px+q, is "x2+px+q completely flat?" .

mark

1. Fraction: Generally, two algebraic expressions are represented by A and B, and A÷B can be represented as one form. If b contains letters, the formula is called a fraction.

2. Rational formula: Algebraic formula and fractional formula are collectively called rational formula; Namely.

3. Two important judgments about the score: (1) If the denominator of the score is zero, the score is meaningless, and vice versa; (2) If the numerator of a fraction is zero and the denominator is not zero, the value of the fraction is zero; Note: If the numerator of a fraction is zero and the denominator is zero, the fraction is meaningless.

4. The basic nature and application of scores;

(1) If both the numerator and denominator of a fraction are multiplied (or divided) by the same non-zero algebraic expression, the value of the fraction remains unchanged;

(2) Note: In a fraction, if any two symbols of numerator, denominator and fraction itself are changed, the value of the fraction remains unchanged;

that is

(3) When simplifying the complex fraction, it is relatively simple to multiply the numerator denominator by the least common multiple of the decimal denominator.

5. Fraction: The divisor of the numerator and denominator of a fraction is called a fraction; Note: Factorization is often needed before fractional reduction.

6. simplest fraction: There is no common factor between the numerator and denominator of a fraction. This fraction is called simplest fraction; Note: The final result of score calculation requires simplification to the simplest score.

7. The law of multiplication and division of scores:

8. the power of the score:.

9. The calculation rules of negative integral index:

(1) formula: A0 = 1 (A ≠ 0), a-n = (a ≠ 0);

(2) The algorithm of positive integer exponent can be used to calculate negative integer exponent;

(3) Formula:

(4) Formula: (-1)-2= 1, (-1)-3=- 1.

10. General fraction of fractions: according to the basic properties of fractions, several fractions with different denominators are converted into fractions with the same denominator equal to the original fraction, which is called general fraction of fractions; Note: The simplest common denominator should be determined before the general division of fractions.

1 1. Determination of the simplest common denominator: the least common multiple of the coefficient? The highest power of the same factor.

12. Rules for addition and subtraction of fractions with the same denominator and different denominator:.

13. One-variable linear equation with letter coefficient: In the equation ax+b=0(a≠0), X is unknown, and A and B are known numbers expressed by letters. For X, the letter A is the coefficient of X, which is called the letter coefficient, and the letter B is a constant term, which is called a one-dimensional linear equation with letter coefficient. note:

14. Formula deformation: transforming a formula from one form to another is called formula deformation; Note: The essence of formula deformation is to solve equations with letter coefficients. It is particularly important to note that when both sides of the letter equation are multiplied by the algebraic expression with letters at the same time, it is generally necessary to confirm that the value of this algebraic expression is not 0 first.

15. Fractional equation: the equation with unknown number in denominator is called fractional equation; Note: previously learned equations with unknown denominator are integral equations.

16. Rooting of fractional equation: When solving fractional equation, in order to remove the denominator, both sides of the equation have to be multiplied by algebraic expressions containing unknowns, so root growth may occur, so the fractional equation must be tested for root growth; Note: When solving the equation, don't divide both sides of the equation by algebraic expressions containing unknowns at the same time, because the roots may be lost.

17. Method of testing the root of fractional equation: substitute the root of fractional equation into the simplest common denominator (or each denominator of fractional equation). If the value is zero, the root is the root, and then the original equation has no solution; If the value is not zero, the root is the solution of the original equation; Note: From this, it can be judged that the unknown value that makes the denominator zero may be the root of the original equation.

18. Application of fractional equation: Fractional equation can solve application problems in the same way as integral equation, except that the program of "checking and increasing roots" needs to be added.

Root of number

Definition of 1. square root: If x2=a, X is called the square root of A (that is, the square root of A is X); Note: (1)a is the square number of X, (2) It is known that X is a power, A is a root, and the power sum root is a reciprocal operation.

2. The nature of the square root:

The square root of (1) positive number is a pair of opposites;

(2) The square root of 0 is still 0;

(3) Negative numbers have no square root.

3. Representation of square root: The square root of A is expressed as sum. Note: it can be regarded as a number, and it can also be regarded as an operation of opening two times.

4. arithmetic square root: the positive square root of a positive number is called the arithmetic square root of a, which is expressed as. Note: The arithmetic square root of 0 is still 0.

5. Three important non-negative numbers: a2≥0, |a|≥0 and ≥0. Note: The sum of non-negative numbers is 0, which means all of them are 0.

6. Two important formulas:

( 1) ; (a≥0)

(2) .

7. Definition of cube root: If x3=a, then X is called the cube root of A (that is, the cube root of A is X). Note: (1)a is called the cubic number of X; (2) The cubic root of A is expressed as: that is, the cubic power of A is opened.

8. The nature of the cube root:

(1) The cube root of a positive number is a positive number;

(2) The cube root of 0 is still 0;

(3) The cube root of a negative number is a negative number.

9. Characteristics of cube root:

10. Irrational number: Infinitely circulating decimals are called irrational numbers. An inexhaustible number with roots is irrational.

1 1. Real number: rational numbers and irrational numbers are collectively called real numbers.

Classification of real numbers: (1) (2).

13. The properties of the number axis: the points on the number axis correspond to real numbers one by one.

14. Approximation of irrational numbers: if the real number calculation result contains irrational numbers and the topic has no approximation requirements, the result should be expressed by irrational numbers; If the topic requires approximation, the result should be expressed as an approximation of irrational numbers. Note: In the approximate calculation of (1), one more bit should be reserved for the intermediate process; (2) Need to remember: