20 12 summary of detailed knowledge points in each chapter of the seventh grade mathematics volume II of Beijing Normal University Edition. (There should be braces or something)

The knowledge points under the University of North Seven are sorted out in chapters.

Chapter 1: The operation of algebraic expressions.

First of all, the monomial

1, an algebraic expression of the product of numbers and letters, is called a monomial.

2. The single numerical factor is called the single coefficient.

3. The index of all the letters in the monomial and the number of times called the monomial.

4. A single number or letter is also a monomial.

5. The coefficient of monomial with only letter factor is 1 or-1.

6. A number is a monomial, and its coefficient is itself.

7. The degree of a single nonzero constant is 0.

8. A single item can only contain multiplication or power operation, and cannot contain other operations such as addition and subtraction.

9. The coefficient of the monomial includes the symbol before it.

10, when the coefficient of the monomial is a fraction, it should be turned into a false fraction.

When 1 1 and the single coefficient is 1 or-1, the number "1" is usually omitted.

12, the number of monomials is only related to letters, and has nothing to do with the coefficient of monomials.

Second, polynomials

The sum of 1 and several monomials is called a polynomial.

2. Each monomial in a polynomial is called a polynomial term.

3. The term without letters in polynomial is called constant term.

4. A polynomial has several terms, which are called polynomials.

5. Every term of polynomial includes the symbol before the term.

6. Polynomials have no concept of coefficient, but have the concept of degree.

7. The degree of the term with the highest degree in a polynomial is called the degree of the polynomial.

Third, algebraic expressions.

1, monomials and polynomials are collectively called algebraic expressions.

2. Both monomials and polynomials are algebraic expressions.

3. Algebraic expressions are not necessarily monomials.

4. Algebraic expressions are not necessarily polynomials.

5. Algebraic expressions with letters in denominator are not algebraic expressions; It is a fraction to learn in the future.

Fourth, the addition and subtraction of algebraic expressions.

The theoretical basis of 1. Algebraic expression addition and subtraction is: the rule of removing brackets, the rule of merging similar items, and the rule of multiplication and distribution.

2. The key to the addition and subtraction of several algebraic expressions is to use the rule of brackets correctly, and then merge similar items accurately.

3, several general steps of algebraic expression addition and subtraction:

(1) List algebraic expressions: enclose each algebraic expression in parentheses and then connect it with a plus sign and a minus sign.

(2) Open brackets according to the rules for opening brackets.

(3) Merge similar items.

4, the general steps of algebraic evaluation:

(1) algebraic simplification.

(2) Substitution calculation

(3) For some special algebraic expressions, "whole substitution" can be used for calculation.

V. Multiplication with the same base number

1, multiplied by n identical factors (or factors) A, recorded as An, read as the n power (power) of A, where A is the base, n is the exponent, and an.

The result is called power.

2. Powers with the same base are called same base powers.

3. Same base multiplication algorithm: same base multiplication, constant base, exponential addition. Namely: am ﹒ an = am+n

This rule can also be reversed, that is, am+n = am-an.

5. Start the power of different cardinality. If it can be converted into a power with the same base, first turn it into a power with the same base, and then apply the rules.

Sixth, the power of power.

The sum of powers of 1 refers to the multiplication of several identical powers. (am)n represents the multiplication of n am.

2. Power algorithm: power, constant basis, exponential multiplication. (am)n =amn .

3. This rule can also be reversed, that is, AMN = (am) n = (an) m.

Seven, the product of power

1, the power of product is the power of cardinal number and product.

2. Multiplication algorithm of product: Multiplication of product is equal to multiplying each factor in the product separately, and then multiplying the obtained power.

3. This rule can also be reversed, namely: AnBN = (AB) n. VIII. Similarities and differences of three kinds of "power arithmetic" 1, * * * Similarities and differences:

The cardinality in the (1) rule remains unchanged, and only the exponent is operated.

(2) The cardinal number (non-zero) and exponent in the law are universal, that is, they can be numbers or formulas (single or multiple terms).

(3) For operations with three or more operations, the rule still holds.

2. Differences:

(1) The same base power multiplication is exponential addition.

(2) The power of the power is exponential multiplication.

(3) The product is multiplied by each factor and then multiplied by the result.

Nine, the same base power division

1, same base powers's division rule: same base powers division, base constant, minus the exponent, that is, am÷an = am-n.

(a≠0).

2. this rule can also be used in reverse, that is, am-n = am÷an.

(a≠0). Ten, zero exponential power

The meaning of 1 and zero exponential power: the power of 0 of any number that is not equal to 0 is equal to 1, that is, a0.

= 1(a≠0).XI。 Negative exponential power

1, the -p power of any number that is not equal to zero is equal to the reciprocal of the p power of this number, that is,1(0) pp.

aa？

Note: In same base powers's division, the base of zero exponential power and negative exponential power is not 0. XII. Multiplication of Algebraic Expressions

(1) Multiplies the monomial by the monomial.

1, the rule of monomial multiplication: the monomial is multiplied by the monomial, and their coefficients are multiplied by the power of the same letter, respectively, and the remaining letters, together with their exponents, remain unchanged as the factors of the product. 2, coefficient multiplication, pay attention to symbols.

3. The powers of the same letters are multiplied, the base is unchanged, and the exponents are added.

4. For the letters only contained in the monomial, write them together with its index as the factor of the product. 5. The result of multiplying the monomial by the monomial is still the monomial.

6. The multiplication rule of monomials also applies to the multiplication of three or more monomials. (2) Multiplication of monomial and polynomial

1. Multiplication rule of monomial and polynomial: Multiplying monomial and polynomial means multiplying each term in polynomial by monomial according to distribution rate, and then adding the products. Namely: m(a+b+c)=ma+mb+mc. 2. Please pay attention to the product logo when operating. Every term of a polynomial is preceded by a symbol. 3. The product is a polynomial with the same number of terms as the polynomial.

4. When mixing operations, pay attention to the operation sequence. If there are similar items in the results, they should be merged to get the simplest result. (3) Multiplication of Polynomials and Polynomials

1, polynomial and polynomial multiplication rule: polynomial multiplication, first multiply each term of a polynomial with another term.

For each term in the formula, add the products. That is: (m+n)(a+b)=ma+mb+na+nb.

2. Polynomial multiplication must not be repeated or missed. Multiplication should be carried out in a certain order, that is, every term of one polynomial should be multiplied by every term of another polynomial. Before merging similar terms, the number of terms of the product is equal to the product of two polynomial terms.

3. Every term of a polynomial is preceded by a symbol. When determining the symbols of terms in products, "the same number is positive and the different number is negative" should be applied.

4. If there are similar items in the operation results, they should be merged.

5. For the multiplication of two linear binomials of the same letter with a linear coefficient of 1, the following formula can be used.

Simplified operation: (x+a)(x+b)=x2.

+(a+b)x+ab. XIII. Square difference formula

1 、( a+b)(a-b)=a2-b2

That is to say, the product of the sum of two numbers and the difference between them is equal to the difference of their squares. 2. A and B in the square difference formula can be monomials or polynomials.

3. The square difference formula can be reversed, namely a2-b2.

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

4. The square difference formula can also simplify the operation of the product of two numbers. To solve this kind of problem, we must first see whether two numbers can be converted into

(a+b)？ (a-b), then look at a2 and b2.

Whether it is easy to calculate. Fourteen, the complete square formula

1、222222

() 2, () 2, abaabbabaabb, that is, the square of the sum (or difference) of two numbers is equal to the sum of their squares, plus (or minus) twice their product. 2. A and B in the formula can be monomials or polynomials. 3. Master and understand the deformation formula of the complete square formula:

( 1)222222

()2()2[()()]abababab？ (2)22

()4ababab

(3)22

14[()()]ababab 4, completely flat mode: we put the shape as follows: 2222.

2, 2, aabbaabb's quadratic trinomial is called completely flat mode.

5. When calculating the square of a large number, you can use the complete square formula to simplify the operation of the number.

6. The complete square formula can be reversed, that is, 222222.

2 (), 2 (). aabbbabababab fifteen. Division of algebraic expressions

(a) the law of dividing the monomial by the monomial

1, the law that the monomial is divided by the monomial: Generally speaking, when the monomial is divided, the coefficient and same base powers are separated as factors of quotient; For the letter only contained in the division formula, it is used as the factor of quotient together with its index. 2. According to the law, the calculation method of monomial division is similar to that of monomial multiplication, and it is also divided into three parts: coefficient, same letter and different letter. (b) The rule of polynomial divided by monomial

1, the law of polynomial divided by monomial: polynomial divided by monomial, first divide each term of this polynomial by monomial, and then add the obtained quotients. Expressed in letters: (). abcmambmcm？ 2. Divide the polynomial by the monomial, and note that each term of the polynomial includes the previous symbol.

That is, (ab)n=anbn.

3. This rule can also be reversed, that is, AnBN = (ab) n.

. Eight, the similarities and differences of three kinds of "power arithmetic" 1, * * * similarities:

The cardinality in the (1) rule remains unchanged, and only the exponent is operated.

(2) The base (nonzero) and exponent in the law are universal, that is, they can be numbers or formulas (monomials or polynomials).

(3) For operations with three or more operations, the rule still holds. 2. Differences:

(1) The same base power multiplication is exponential addition. (2) The power of the power is exponential multiplication.

(3) The product is multiplied by each factor and then multiplied by the result. Nine, the same base power division