Mathematics eighth grade volume 2 knowledge 1
One-dimensional linear inequality and one-dimensional linear inequality system
I. Unequal relations
1. Generally, formulas connected by symbols ""(or "≥") are called inequalities.
2. Accurately "translate" inequalities and correctly understand mathematical terms such as "non-negative number" and "not less than". ※ 。
nonnegative number
Nonpositive number
Second, the basic properties of inequality
1. Master the basic properties of inequalities and use them flexibly. ※:
Add (or subtract) the same algebraic expression on both sides of inequality (1), and the direction of inequality remains unchanged, namely:
If a>b, then A+C > b+c，a-c & gt； b-c。
(2) Both sides of the inequality are multiplied by (or divided by) the same positive number, and the direction of the inequality sign remains the same, that is:
If a>b and c>0, then ac> BC,
(3) When both sides of the inequality are multiplied by (or divided by) the same negative number, the direction of the inequality changes, namely:
If a>b and c < 0, AC
2. Comparison size: (A and B represent two real numbers or algebraic expressions respectively). ※
Generally speaking:
If a>b, then a-b is a positive number; On the other hand, if a-b is positive, then a >；; b；
If a=b, then a-b is equal to 0; On the other hand, if a = b；; Is equal to 0, then a = b；;
If a
Namely:
a & gtb & lt= = = & gta-b & gt； 0
a = b & lt= = = & gta-b=0
a a-b & lt； 0
3. Solution set of inequality;
1. The value of the unknown quantity that can make the inequality hold is called the solution of the inequality. All the solutions of an inequality constitute the solution set of this inequality. ※: The process of finding the solution set of inequality is called solving inequality.
2. There are countless solutions to inequality, generally all numbers in a certain range, which are different from the solutions of equations. ※
3. Representation of inequality solution set on the number axis;
When using the number axis to represent the solution set of inequality, we should determine the boundary and direction:
① Boundary: a solid circle with an equal sign and a hollow circle without an equal sign;
② Direction: large on the right and small on the left.
4. One-dimensional linear inequality;
1. A formula containing only one unknown is an algebraic expression, and the degree of the unknown is 1. Inequalities like this are called unary linear inequalities. ※.
2. The process of solving one-dimensional linear inequality is similar to solving one-dimensional linear equation. When both sides of the inequality are multiplied by negative numbers, the sign of the inequality will change direction. ※.
3. Steps to solve linear inequality of one variable. ※:
1 naming;
(2) the bracket is removed;
③ shifting items;
(4) merging similar projects;
⑤ The coefficient is changed to 1 (the problem of changing inequality)?
4. The basic situation of one-dimensional linear inequality is ax > ※ b (or axe.
① when a >; 0, the solution is;
② when a=0, b
When a=0 and b≥0, there is no solution;
③ when a
5. The basic steps of solving application problems with column inequalities are similar to those of solving application problems with column equations, namely:
(1) Examination: Carefully examine the questions, find out the unequal relations in the questions, and grasp the key words in the questions, such as "greater than", "less than", "not greater than" and "not less than";
(2) setting: setting appropriate unknowns;
③ Column: list inequalities according to the inequality relations in the question;
④ Solution: Solve the solution set of the listed inequalities;
⑤ Answer: Write the answer and check whether the answer conforms to the meaning of the question.
One-dimensional linear inequality system of intransitive verbs
1. Definition: An inequality group consisting of several linear inequalities with the same unknown number is called a linear inequality group. ※.
2. The common part of each inequality solution set in one-dimensional linear inequality group is called the solution set of inequality group. If the solution set of these inequalities has no common part, it is said that this inequality set has no solution. ※. (The common part of the solution set is usually determined by the number axis. )?
3. Steps to solve linear inequalities. ※:
(1) Find the solution set of each inequality in the inequality group;
(2) Use the number axis to find the common part of these solution sets, that is, the solution set of this inequality group.
Four cases of the solution set of two linear inequalities (A and B are real numbers, A
X> is the larger of the two.
X>a, whichever is the smallest.
A<x<b, find the middle of the big and small crosses.
No solution, size separation no solution (empty set)
Mathematics eighth grade, Volume II, Knowledge II
Translation and rotation of graphics
I. Translation transformation:?
1. Concept: In a plane, a graphic moves a certain distance along a certain direction, and such graphic movement is called translation. ?
2. Nature:
(1) graphic congruence before and after translation; ?
(2) The connecting lines of corresponding points are parallel or on the same straight line and equal.
3. Drawing steps and translation methods:?
(1) Distinguish the topic requirements and determine the direction and distance of translation;
(2) Analyze the graph to find out the key points that constitute the graph;
(3) translating each key point along a certain direction by a certain distance;
(4) Connect the key points and mark them with corresponding letters;
(5) write a conclusion. ?
Second, the rotation transformation:?
1. Concept:
In a plane, turning a figure around a fixed point by an angle in a certain direction is called rotation. ?
Description:
The rotation of (1) graph is determined by the rotation center and rotation angle;
(2) In the process of rotation, the center of rotation is always fixed.
(3) In the process of rotation, the rotation direction is consistent.
(4) When the rotation process is stable, the rotation angle of a point on the diagram is the same.
Rotation does not change the size and shape of the graph.
2. Nature:
(1) The distance from the corresponding point to the rotation center is equal; ?
(2) The included angle of the connecting line between the corresponding point and the rotation center is equal to the rotation angle;
(3) Graphic congruence before and after rotation.
3. Steps and methods of rotating drawing:
(1) Determine the rotation center, rotation direction and rotation angle;
(2) Find out the key points of the graph;
(3) connecting the key points of the graph with the rotation center, and then rotating a rotation angle according to the rotation direction to obtain the corresponding points of these key points;
(4) Connect these corresponding points in turn according to the original image, and the resulting graph is the rotated graph.
Note: When drawing by rotation, the included angle between a pair of corresponding points and the rotation center is the rotation angle.
4. Common test methods?
(1) proves triangle congruence by combining translation and rotation;
(2) Using the properties of translation transformation and rotation transformation, some topics are designed.
Mathematics eighth grade, Volume II, Knowledge 3
factoring
I. Factorization
1. Converting a polynomial into the product of several algebraic expressions is called decomposing this polynomial. ※.
2. Factorization and algebraic expression multiplication are reciprocal. ※:
Differences and relations between factorization and algebraic expression multiplication;
(1) Algebraic expression multiplication is to multiply several algebraic expressions into polynomials;
(2) Factorization is to multiply a polynomial by several factors.
Two. Improve the public's factorial method.
1. If each term of a polynomial contains a common factor, then this common factor can be proposed, so that the polynomial can be transformed into the product of two factors. This factorization method is called extraction of common factors. ※.
※ 2. Concept connotation:
The final result of factorization of (1) should be "product";
(2) The common factor can be a monomial or polynomial;
(3) The theoretical basis of common factor method is the distribution law of multiplication to addition.
3.※ Comments on error-prone points:
(1) Pay attention to whether the sign of the power exponent term is wrong;
(2) Whether the common factor formula is "clean";
(3) One of the polynomials is only a common factor; After being put forward; This item in brackets is+1; Don't miss it.
Three. Formula method
1. If the multiplication formula is reversed, it can be used to factorize some polynomials. This factorization method is called formula method. ※.
※ 2. Main formula:
(1) square difference formula: a2-b2=(a+b)(a-b)
(2) Complete square formula: picture
3. Use the formula method. ※:
(1) square difference formula: a2-b2=(a+b)(a-b)
(1) Binomial or polynomial should be regarded as binomial;
(2) Every term of binomial (unsigned) is the square of monomial (or polynomial);
Binomials are different symbols.
(2) Complete square formula: picture
(1) should be a trinomial;
(2) where two numbers are the same and each is the square of an algebraic expression;
(3) There is another term that can be positive or negative, and it is twice the base product of the first two terms.
4. Thinking and solving steps of factorization. ※:
(1) First check whether each item has a common factor, and if so, extract the common factor first;
(2) See if the formula method can be used;
(3) Using the grouping decomposition method, that is, extracting the common factors of each group after grouping or using the formula method to achieve the purpose of decomposition;
(4) The final result of factorization must be the product of several algebraic expressions, otherwise it is not factorization;
(5) The results of factorization must be carried out until each factorization can no longer be decomposed within the scope of rational numbers.
4. Grouping decomposition method:
1. Grouping decomposition method: The method of grouping factors is called grouping decomposition method. ※.
draw
※ 2. Concept connotation:
The key of grouping decomposition method is how to group, whether there are common factors to be extracted after grouping, whether it can continue to decompose, and whether it can continue to decompose factors by formula after grouping.
3. Note: Pay attention to the change of symbols when grouping. ※.
Verb (abbreviation for verb) cross multiplication:
1. For quadratic trinomial pictures, a and c are decomposed into the products of two factors respectively. ※? Pictures and satisfying pictures are often written in the form of pictures and decomposed by quadratic trinomial.
2. Decomposition of quadratic trinomial graph. ※:
Pictures?
※ 3. Connotation of law:
(1) Understanding: In factorization, if the constant term q is positive, it is decomposed into two factors with the same sign, and their signs are the same as those of the coefficient p of the first term.
(2) If the constant term Q is negative, it is decomposed into two factors with different signs, in which the factor with larger absolute value has the same sign as the linear term coefficient P. For the decomposed two factors, it depends on whether their sum is equal to the linear term coefficient P. ..
4. Error-prone notes:
(1) cross multiplication is easy to make mistakes when decomposing coefficients;
(2) The result of decomposition is different from the original formula. At this time, polynomial multiplication is usually used to test whether the decomposition is correct.
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