1, algebraic expression
Expressions that connect numbers or letters representing numbers with operational symbols are called algebraic expressions. A single number or letter is also algebraic.
2. Single item
Algebraic expressions that contain only the product of numbers and letters are called monomials.
Note: The monomial is composed of coefficients, letters and indices of letters, in which the coefficients cannot be expressed by fractions. For example, this expression is wrong and should be written as. In a monomial, the sum of the exponents of all the letters is called the degree of the monomial. If it's a 6-degree monomial. Test point 2, polynomial (1 1)
1, polynomial
The sum of several monomials is called polynomial. Where each monomial is called a term of this polynomial. Items without letters in polynomials are called constant terms. The degree of the term with the highest degree in a polynomial is called the degree of the polynomial.
Monomial and polynomial are collectively called algebraic expressions.
Replacing the letters in the algebraic expression with numerical values and calculating the results according to the operations specified in the algebraic expression are called algebraic values.
Note: (1) To find the value of the algebraic expression, it is generally to simplify the algebraic expression first and then substitute the value of the letter.
(2) To find an algebraic value, sometimes we can't find the value of its letters, so we need to use skills and "whole" substitution.
2. Similar projects
Items with the same letter and the same letter index are called similar items. Several constant terms are similar.
3. Rules for removing brackets
(1) brackets are preceded by "+". Remove the brackets together with the preceding "+"sign, and all items in the brackets remain unchanged.
(2) There is a "-"before the brackets. Remove the brackets together with the "-"in front, and all the items in the brackets have changed.
4. Algebraic expression algorithm
Addition and subtraction of algebraic expressions: (1) bracket removal; (2) Merge similar items. Note: (1) The result of single item multiplied by single item is still single item.
(2) Multiply the monomial with the polynomial to get a polynomial with the same number of terms as the polynomial in the factor.
(3) Pay attention to the symbol problem when calculating. Each term of a polynomial contains the symbol before it, and we should also pay attention to the symbol of a single term.
(4) In the expansion of polynomial multiplication, if there are similar items, they should be merged.
(5) The letters in the formula can represent numbers, monomials or polynomials.
(6) Divide the polynomial by the monomial, first divide each term of the polynomial by the monomial, and then add the obtained quotients. The division of a monomial by a polynomial cannot be calculated in this way.
After the senior high school entrance examination every year, the most talked about by teachers and students is the difficulty of mathematics and geometry in the senior high school entrance examination. In a sense, how well the senior high school entrance examination geometry is done directly determines whether the senior high school entrance examination mathematics can get high marks. From this point of view, geometry in mathematics is very important for mathematics in senior high school entrance examination. Those who get geometry get math in the middle school entrance examination.
Usually, the geometry of the senior high school entrance examination is presented as follows: the small questions in the multiple-choice questions calculate the corresponding angles and line segments, and the corresponding calculations are also based on the fill-in-the-blank questions. Choose four points to fill in the blanks for each question.
Next, in solving problems, we will usually examine the 22nd question of simple congruent triangles, proof of tangent in circle and proof of calculation in circle, hands-on operation or geometric flexible thinking ability, 24th question of substituting several comprehensive questions and 25th question of geometric comprehensive finale. Among them, questions 22, 24 and 25 are usually called the final questions of mathematics in the senior high school entrance examination, and whether these three questions are done well is directly related to the level of mathematics scores in the senior high school entrance examination.
1, pay attention to the foundation in the new curriculum. When learning new courses at school, we must lay a good foundation and make every basic knowledge point clear. Understand every theorem and the proof method of theorem, so as to associate with relevant knowledge points. Always take notes in class and remember every thought that flashes.
2. Pay attention to induction. Summarize the related questions in the textbook guidance books that you have done, review them frequently, and mark the important questions at the same time.
3. Maintain the proficiency in adding auxiliary lines in quadrangles and triangles. Especially the three transformations of geometry, rotation, translation and axial symmetry, we should be skilled and practice this kind of topic more.
4. Practice more questions.
5. Master the mathematical model of junior high school. Master the model and skillfully use the ladder skills.
I. Key concepts
Classification:
1。 Algebraic and rational expressions
Formulas that associate numbers or letters representing numbers with operational symbols are called algebraic expressions. independent
Numbers or letters are also algebraic.
Algebraic expressions and fractions are collectively called rational forms.
2。 Algebraic expressions and fractions
Algebraic expressions involving addition, subtraction, multiplication, division and multiplication are called rational expressions.
Rational expressions without division or division but without letters are called algebraic expressions.
Rational number formula has division, and there are letters in division, which is called fraction.
3。 Monomial and polynomial
Algebraic expressions without addition and subtraction are called monomials. (product of numbers and letters-including single numbers or letters)
The sum of several monomials is called polynomial.
Note: ① According to whether there are letters in the division formula, algebraic expressions and fractions are distinguished; According to whether there are addition and subtraction operations in algebraic expressions, monomial and polynomial can be distinguished. ② When classifying algebraic expressions, the given algebraic expressions are taken as the object, not the deformed algebraic expressions. When we divide the category of algebra, we start from the representation. For example,
=x, =│x│ and so on.
4。 Coefficient and index
Difference and connection: ① from the position; (2) In the sense of representation.
5。 Similar projects and their combinations
Conditions: ① The letters are the same; ② The indexes of the same letters are the same.
Basis of merger: law of multiplication and distribution
6。 radical expression
The algebraic expression of square root is called radical.
Algebraic expressions that involve square root operations on letters are called irrational expressions.
Note: ① Judging from the appearance; ② Difference: It is a radical, but it is not an irrational number (it is an irrational number).
7。 arithmetic square root
(1) The positive square root of a positive number ([the difference between a ≥ 0-and "square root"]);
⑵ Arithmetic square root and absolute value
① Contact: all are non-negative, =│a│.
② Difference: │a│, where A is all real numbers; Where a is a non-negative number.
8。 After the same quadratic root, the simplest quadratic root and denominator are materialized into the simplest quadratic root, the quadratic roots with the same number of roots are called the same quadratic root.
The following conditions are satisfied: ① the factor of the root sign is an integer and the factor is an algebraic expression; (2) The number of roots does not include exhausted factors or factors.
Crossing out the root sign in the denominator is called denominator rationalization.
9。 index
(1)(- power supply, power supply operation)
(1) when a > 0, > 0; ② when a < 0, > 0 (n is even) and < 0 (n is odd)
(2) Zero index: = 1(a≠0)
Negative integer index: = 1/ (a≠0, p is a positive integer)
Second, the law of operation and the law of nature
1。 Rules of addition, subtraction, multiplication, division, multiplication and root of fractions
2。 Properties of fractions
(1) Basic properties: = (m≠0)
(2) Symbolic law:
⑶ Complex fraction: ① Definition; ② Simplified methods (two kinds)
3。 Algebraic expression algorithm (rules of removing brackets and adding brackets)
4。 The operational attributes of power: ①? = ; ② ÷ = ; ③ = ; ④ = ; ⑤
Skills:
5。 Multiplication rule: (1) single× single; (2) single × many; 3 more x more.
6。 Multiplication formula: (plus or minus)
(a+b)(a-b)= 1
(a b) =
7。 Division rule: (1) single-single; (2) Too many orders.
8。 Factorization: (1) definition; ⑵ Methods: A. Common factor method; B. formula method; C. cross multiplication; D. group decomposition method; E. root formula method.
9。 Properties of arithmetic roots: =; ; (a≥0，b≥0)； (a≥0, b > 0) (positive and negative)
10。 Radical operation rule: ⑴ addition rule (combining similar quadratic roots); (2) multiplication and division; (3) The denominator is reasonable: a; b； c .
1 1。 Scientific notation: (1 ≤ A < 10, n is an integer =
Third, the application examples (omitted)
Four, comprehensive operands (omitted)