(a) using the formula method:

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:

a2-b2=(a+b)(a-b)

a2+2ab+b2=(a+b)2

a2-2ab+b2=(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): a2-b2=(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) Reversing the multiplication formula (a+b)2=a2+2ab+b2 and (a-b)2=a2-2ab+b2, we can get:

a2+2ab+b2 =(a+b)2

a2-2ab+b2 =(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.

Equations a2+2ab+b2 and a2-2ab+b2 are called completely flat modes.

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. Use the formula x2 +(p+q)x+pq=(x+q)(x+p) for factorization, and pay attention to:

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 product form of the factors, and then we can omit the common factor of the numerator and denominator. If the polynomial in the numerator or denominator can't decompose the factor, we can't separate some items in the numerator and denominator at this time.

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.

(8) 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.

Key points of eighth grade mathematics chapter knowledge

Chapter 17 points for reviewing scores

1, the formula with the shape of AB(A and B are algebraic expressions, B contains letters, and B≠0) is called a fraction. Algebraic expressions and fractions are collectively called rational forms.

2. When the denominator is ≠0, the score is meaningful. When the denominator = 0, the score is meaningless.

3. When the score is 0, two conditions must be met at the same time: numerator = 0 and denominator ≠0.

4. Basic properties of the fraction: both the numerator and denominator of the fraction are multiplied or divided by the same algebraic expression that is not 0, and the value of the fraction remains unchanged.

5. The signs of the fraction, numerator and denominator can be changed at will, and the value of the fraction remains unchanged.

6. Four Fractional Operations

1) The key to the addition and subtraction of fractions is general division. Fractions with different denominators are transformed into fractions with the same mother, and then operations are performed.

2) When multiplying and dividing fractions, factorize the numerator and denominator first, and then omit the same factor.

3) the mixed operation of fractions, pay attention to the change of operation order and sign,

4) The final result of fractional operation should be reduced to the simplest fractional or algebraic expression.

7. Fractional equation

1) fractional simplification cannot be confused with solving fractional equations. Fractional simplification is an identical deformation, and the denominator cannot be removed at will.

2) Steps of solving the fractional equation: firstly, the fractional equation is transformed into an integral equation; Second, solve the whole equation; Third, check the roots and remove the added roots through inspection.

3) The steps of solving related application problems are the same as those of listing integral equations to solve application problems: setting, listing, solving, testing and answering.

Chapter 18 review points of functions and images

1, and the straight line specifying the origin, positive direction and unit length is called the number axis. Points on the number axis correspond to real numbers one by one. If the coordinates of point A and point B on the number axis are x 1 and x2, then AB =.

2. Two number axes have a common origin and are perpendicular to each other, forming a plane rectangular coordinate system. Points on the coordinate plane correspond to ordered real number pairs one by one.

3. The points on the coordinate axis do not belong to any quadrant. The ordinate of this point on the x axis y = 0；; The abscissa of a point on the y axis x = 0.

Point x > in the first quadrant; 0，y & gt0; Point x in the second quadrant

Therefore, for the point above the X axis, the ordinate y > 0；; The point below the X axis, the ordinate y < 0；; The point on the left side of the Y axis, the abscissa x < 0；; The point on the right side of the y axis, the abscissa x > 0.

4. For a point symmetrical about a coordinate, the coordinate of this axis is unchanged, and the coordinate of the other axis is opposite. For a point with symmetrical origin, the vertical axis and the horizontal axis are opposite. About the point where the bisector of the first quadrant and the third quadrant are symmetrical, the abscissa and ordinate are interchanged; With regard to the symmetrical points on the bisector of the second quadrant and the fourth quadrant, not only the abscissa and the ordinate exchange positions, but also they become opposite numbers.

5. The horizontal and vertical coordinates of the points on the bisector of the first quadrant and the third quadrant are equal; The horizontal and vertical coordinates of the points on the bisector of the second and fourth quadrants are opposite to each other.

6. In a changing process, there are two variables X and Y. For each value of X, Y has a unique value corresponding to it, so we say that Y is a function of X. X is an independent variable and Y is a dependent variable. The expression methods of functions are: analytical method, image method and list method.

7. The range of independent variables of the function: ① When the analytic expression of the function is algebraic expression, all independent variables can be real numbers; (2) When the analytic expression of the function is a fraction, the value of the independent variable should make the denominator ≠ 0; ③ When the analytic formula of the function is a quadratic root, the value of the independent variable should make the root sign ≥ 0. (4) When the analytic expressions of the function are negative integer exponent and zero exponent, the cardinality is ≥ 0; ⑤ To embody the functional relationship of practical problems and make practical problems meaningful.

8. if y = kx+b (k and b are constants, k≠0), then y is called a linear function of x, if y = kx (k is a constant, k 0), then y is a proportional function of X.

9. The algebraic meaning of a point on a function image is that the coordinates of the point satisfy the analytical formula of the function. The algebraic significance of the intersection of two functions is that the solution of the equations formed by the analytical expressions of the two functions is the intersection coordinates.

10 and the properties of linear function y = kx+b;

The image of (1) linear function is a straight line passing through two points. The larger the value of |k|, the closer the image is to the y axis.

(2) when k >; 0, the image passes through one or three quadrants, and y increases with the increase of x; From left to right, the image is rising (lower left and higher right);

(3) When k < 0, the image passes through two or four quadrants, and y decreases with the increase of x, and from left to right, the image decreases (the left is higher and the right is lower);

(4) When b>0, the intersection point (0, b) with the Y axis is on the positive semi-axis; When b<0, the intersection (0, b) with the Y axis is on the negative semi-axis. When b = 0, the linear function is a proportional function and the image is a straight line passing through the origin.

(5) When several straight lines are parallel to each other, the values of k are equal and b are not equal.

1 1, if y = kx (k is constant, k≠0), then y is called the inverse proportional function of x.

12, the property of inverse proportional function y = kx;

(1) The image of the inverse proportional function is a hyperbola, which is infinitely close to the X and Y axes.

(2) when k >; 0, the two branches of the image are located in the first and third quadrants. In each quadrant, y decreases with the increase of x, and the image decreases from left to right (lower left and upper right).

(3) When k < 0, the two branches of the image are located in the second and fourth quadrants. In each quadrant, y increases with the increase of x, and the image rises from left to right (high left and low right).

(4) The intersection of inverse proportional function y = kx and positive proportional function y = kx is symmetrical about the origin.

Chapter 19 congruent triangles

1. A sentence that judges right or wrong is called a proposition. A correct proposition is called a true proposition, and a wrong proposition is called a false proposition.

2. The proposition consists of two parts: the topic and the conclusion. The topic is what is known; A conclusion is something deduced from what is known. It can often be written in the form of "If …………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………… The part that begins with "if" is the title, and the part that begins with "then" is the conclusion.

3. The two acute angles of a right triangle are complementary.

4, triangle congruence judgment:

Method 1: If two triangles have two sides and their included angles are equal, then the two triangles are congruent. The abbreviation is S.A.S (or angular).

Method 2: If two triangles have two angles and their clamping edges are equal, then the two triangles are congruent. The abbreviation is A.S.A (or corners).

Method 3: If two triangles have two angles and the opposite sides of one angle are equal, then the two triangles are congruent. The abbreviation is A.A.S (or corner edge).

Method 4: If the three sides of two triangles correspond equally, then the two triangles are congruent. The abbreviation is S.S.S (or edge to edge).

Method 5 (only applicable to right-angled triangles): If the hypotenuse and one right-angled side of two right-angled triangles are equal respectively, then the two right-angled triangles are identical. The abbreviation is H.L. (or hypotenuse and right angle).

Generally speaking, in two propositions, if the topic of the first proposition is the conclusion of the second proposition and the conclusion of the first proposition is the topic of the second proposition, then these two propositions are called reciprocal propositions. If one of the propositions is called the original proposition, then the other proposition is called its inverse proposition.

6. If the inverse proposition of a theorem is also a theorem, then these two theorems are called reciprocal theorems, and one of them is called the inverse theorem of the other theorem.

7. If the two angles of a triangle are equal, then the opposite sides of the two angles are also equal.

8. If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then this triangle is a right triangle. (Inverse Theorem of Pythagorean Theorem)

9. The points on the bisector of an angle are equidistant from both sides of the angle. The points with equal distance to both sides of the angle are on the bisector of the angle.

10, the distance between the point on the middle vertical line of the line segment and the two endpoints of this line segment is equal; The point with the same distance to the two endpoints of a line segment is on the middle vertical line of this line segment.

Chapter 20 Determination of Parallelogram

1, quadrilateral interior angle sum theorem: quadrilateral interior angle sum is equal to 360;

2. The theorem of polygon interior angle sum: the sum of n polygon interior angles is equal to (n-2) ×180;

3. Theorem of the sum of external angles of polygons: the sum of external angles of any polygon is equal to 360;

4. The number formula of diagonal lines of n polygons: n (n-3) 2 (n ≥ 3);

5. Center symmetry: rotate the figure around a certain point 180. If it can coincide with another graph, then the two graphs are said to be symmetrical about this point.

6. Centrally symmetric figure: rotate the figure around a certain point 180. If it can coincide with the original figure, then this figure is called a centrosymmetric figure.

7. The essence of central symmetry: the congruence of two figures about central symmetry; For two graphs with symmetrical centers, the connecting line of symmetrical points passes through the symmetrical center and is equally divided by the symmetrical center.

8, the nature and judgment of parallelogram

Category nature judgment

Symmetry of diagonal of side angle

Parallelogram ① Parallel sides ② Equilateral sides ① Diagonal lines are equal.

② The complementary diagonals of adjacent corners are equally divided and the centers are symmetrical. ① Two groups of parallelograms with parallel opposite sides. ② A set of parallelograms with parallel and equal opposite sides. Two groups of parallelograms bisected diagonally are parallelograms.

Rectangle ① The opposite sides are parallel ② The opposite sides are equal, and all four corners are right angles ① The diagonal is bisected.

(2) The diagonal lines are equal, the center is symmetrical and the axis is symmetrical. (1) A parallelogram with right angles is a rectangle.

② A quadrilateral with three right angles and a parallelogram with equal diagonal lines are rectangles.

The rhombus is parallel to the opposite side, the four sides are equal, the diagonal lines are equal, the adjacent angles are complementary, and the diagonal lines are equally divided vertically.

(2) diagonal bisection method; The diagonal center of each group is symmetrical; (1) A set of parallelograms with equal adjacent sides is a diamond.

② A quadrilateral with four equal sides is a rhombus, and a parallelogram with diagonal lines perpendicular to each other is a rhombus.

Square (1) parallel to the opposite side

The four sides are equal and the four corners are right angles.

(2) Diagonal bisector Each group of diagonals is symmetrical. A group of rectangles with equal adjacent sides is a square, a diamond with right angles is a square, and a parallelogram with orthogonal diagonals is a square.

Isosceles trapezoid ① Two bottoms are parallel ② Two waists are equal. The two angles at the same base are equal. Diagonal lines are equal. A trapezoid with two symmetrical waists is an isosceles trapezoid. A trapezoid with two equal angles on the same base is an isosceles trapezoid.

Chapter 2 1 data sorting and preliminary processing

1, average = total amount/total number of copies. The data has only one average value.

Generally speaking, the average value of n numbers is = 1n (x 1+x2+… xn).

Generally speaking, if x 1 appears f 1 time, x2 appears f2 times, xk appears fk times, and F 1+F2+…+FK = n, the average value of these n numbers can be expressed as x = x1f/kloc. Where fin is the weight of xi (I = 1, 2 … k).

Weighted average is another tool to analyze data. When considering different weights, the decision makers' conclusions may change accordingly.

2. Arrange a set of data from small to large (or from large to small) (even if there are equal data, all of them should participate in the arrangement). If the number of data is odd, then the median is the middle data. If the number of data is even, then the median is the average of the two data in the middle. There is only one median in a set of data, which may or may not be one of the data in this set.

3. The data with the highest frequency in a set of data is the mode. A set of data can have multiple patterns or no patterns (when all data in a set of data appear the same number of times, there is no pattern in the set of data).

4. The maximum value minus the minimum value in a set of data is the extreme value range: extreme value range = maximum value-minimum value.

5. We usually use it to represent the variance of a set of data, to represent the average of a set of data, and to represent each original data with,,,, and.

(square unit)

The method of finding variance: first find the average, then find the deviation, then find the sum of the squares of the deviation, and finally find the average.

6. The square root of the obtained variance is the standard deviation.

7. Mean value, range, variance and standard deviation.

A set of data adds or subtracts a number at the same time, the range is unchanged, the average value adds or subtracts this number, the variance is unchanged, and the standard deviation is unchanged.

A set of data is multiplied or divided by a number at the same time, the range and average are multiplied or divided by this number, the variance is multiplied or divided by the square of this number, and the standard deviation is multiplied or divided by this number.

A set of data is multiplied by a number a at the same time, then a number b is added, the range is multiplied or divided by this number a, the average is multiplied or divided by this number a, and b is added, the variance is multiplied by the square of a, and the standard deviation is multiplied by |a|. (Addition and subtraction are not 0)