1. Addition and subtraction of algebraic expressions
1. monomial: a formula that represents the product of numbers or letters. A single number or letter is also called a monomial.
2. Coefficient and frequency of single item: the numerical factor in single item is called the coefficient of single item; The sum of all the letter indices in the monomial is called the number of times of the monomial.
3. Polynomial: The sum of several monomials is called polynomial.
4. Number and degree of polynomials: the number of monomials contained in a polynomial is the number of polynomial terms, and each monomial is called a polynomial term; In polynomial, the degree of the term with the highest degree is called the degree of polynomial.
5. Algebraic expression: ① monomial ② polynomial.
6. Similar items: monomials with the same letters and the same letter index are similar items.
7. Rules for merging similar items: When the coefficients are added, the letter index remains unchanged.
8. Rules for deleting (adding) brackets: When deleting (adding) brackets, if there is a "+"sign before the brackets, all items in the brackets remain unchanged; If there is a "-"before the brackets, all items in the brackets should be changed.
9. Addition and subtraction of algebraic expressions:
Found: (underlined);
2 "+":(be sure to start the merger with a+sign);
Trinity: (merger).
10. Ascending and descending order of polynomials: arranging the terms of a polynomial according to the exponent of a letter from small to large (or from large to small) is called ascending order (or descending order) of this letter.
Second, one-dimensional linear equation
1. Equation: Equations connected by "=" are called equations.
2. The nature of the equation:
Properties of equation 1: Add (or subtract) the same number or the same algebraic expression on both sides of the equation, and the result is still an equation;
Property 2 of the equation: both sides of the equation are multiplied (or divided) by the same non-zero number, and the result is still an equation.
3. Equation: An equation with an unknown number is called an equation.
4. Solution of the equation: the value of the unknown quantity that makes the left and right sides of the equation equal is called the solution of the equation;
Note: "The solution of the equation can be substituted".
5. Moving term: after changing the sign, moving the term of the equation from one side to the other is called moving term. The shift term is based on the equality attribute 1.
6. One-dimensional linear equation: An integral equation with only one unknown number, degree 1 and non-zero coefficient is a one-dimensional linear equation.
7. The standard form of one-dimensional linear equation: ax+b=0(x is unknown, a and b are known numbers, a≠0).
8. General steps for solving one-dimensional linear equations:
Simplify the equation-the basic properties of fraction.
Denominator-the simplest common denominator of the same multiplication (multiplication is not omitted).
Remove the brackets and pay attention to the change of symbols.
Move the item-change the flag (keep in front).
Merge similar items-merged symbols.
The coefficient is1-except before.
9. List a linear equation of one variable to solve application problems:
(1) reading analysis method: reading analysis method
Read the stem carefully, find out the key words that express the equal relationship, such as "big, small, many, few, yes, * * *, combination, right, completion, increase, decrease, match-",list the literal equations with these key words, and set the unknown number according to the meaning of the question. Finally, using the relationship between quantity and quantity in the question, fill in the algebraic expression and get the equations.
(2) Drawing analysis method
Analyzing mathematical problems with graphics is the embodiment of the combination of numbers and shapes in mathematics. Read the question carefully, and draw the relevant figures according to the meaning of the question, so that each part of the figure has a specific meaning. Finding the equation relationship through graph is the key to solve the problem, so as to obtain the basis of concise equation. Finally, using the relationship between quantity and quantity (unknown quantity can be regarded as known quantity), filling in the relevant algebraic expression is the basis of getting the equation.
Third, absolute value.
Geometric definition of 1. absolute value: The distance from the point representing the number A on the general number axis to the origin is called the absolute value of A, and it is recorded as |a|.
2. Algebraic definition of absolute value
(1) The absolute value of a positive number is itself;
(2) The absolute value of a negative number is its inverse;
(3) The absolute value of 0 is 0.
3, which can be expressed in letters as follows
(1) If A >；; 0, then | a | = a
(2) If
(3) If a=0, then |a|=0.
4. It can be summarized as follows
( 1)a≥0，& lt═>； |a|=a (the absolute value of a non-negative number is equal to itself; A number whose absolute value is equal to itself is nonnegative. )
(2)a≤0，& lt═>； |a|=-a (the absolute value of a non-positive number is equal to its inverse; A number whose absolute value is equal to its opposite number is not a positive number. )
5, the nature of the absolute value
The absolute value of any rational number is non-negative, that is, the absolute value is non-negative. Therefore, if a takes any rational number, there is |a|≥0. that is
The absolute value of (1)0 is 0; A number with an absolute value of 0 is 0. That is, a = 0.
(2) The absolute value of a number is non-negative, and the number with the smallest absolute value is 0. That is | a | ≥ 0;
(3) The absolute value of any number is not less than the original number. Namely: | a | ≥ a;
(4) The absolute values of two numbers are the same positive number, and they are opposite. That is, if | x | = a(a >；; 0), then x = a；;
(5) The absolute values of two opposite numbers are equal. That is: |-a|=|a| or | a | = | b | If a+b = 0;
(6) Two numbers with equal absolute values are equal or opposite. That is: |a|=|b|, then a=b or a =-b;
(7) If the sum of absolute values of several numbers is equal to 0, then these numbers are simultaneously 0. That is |a|+|b|=0, then a=0 and b=0. Common properties of non-negative numbers: if the sum of several non-negative numbers is 0, then only these non-negative numbers are 0 at the same time.
6. Comparison of rational numbers
(1) Compare the sizes of two numbers with the number axis: compared with two numbers on the number axis, the one on the left is always smaller than the one on the right;
(2) Compare the sizes of two negative numbers with absolute values: two negative numbers compare the sizes, and the larger absolute value is smaller; Compare the sizes of two numbers with different signs, and the positive number is greater than the negative number.
Fourth, algebraic expressions.
1. algebraic expression: the expression formed by connecting numbers and letters with basic operation symbols is called algebraic expression, such as n,-1, 2n+500, abc. A single number or letter is also algebraic.
2. Monomial: The algebraic expression representing the product of numbers and letters is called monomial. A single number or letter is also algebraic.
3. Coefficient of single item: numerical factor in single item.
4. The number of monomials: the sum of the indices of all the letters in a monomial.
5. Polynomials:
The sum of several monomials is called polynomial. Each monomial is called a polynomial term, and the term without letters is called a constant term.
The degree of the highest degree term in a polynomial is called the degree of the polynomial. The degree of the constant term is 0.
6. Algebraic expression:
Monomial and polynomial are collectively called algebraic expressions.
Note: It is not an algebraic expression whose denominator contains letters.
7, algebraic writing specification:
(1) Numbers and letters, the multiplication sign in letters and letters can be omitted or indicated by "",and numbers can be placed before letters;
(2) When division occurs, it is expressed by a fraction;
(3) When the band score is multiplied by letters, the band score should become a false score;
(4) If the operation result is an addition and subtraction formula, when there is a unit behind it, the whole formula should be enclosed in brackets.