Summary of mathematical formula method in the second volume of the second day of junior high school
(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 still 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).
Summary of mathematical formula method in the second volume of the second day of junior high school
1, there is only one straight line between two points.
2. The line segment between two points is the shortest.
3. The complementary angles of the same angle or equal angle are equal.
4. The complementary angles of the same angle or equal angle are equal.
5. There is one and only one straight line perpendicular to the known straight line.
6. Of all the line segments connecting a point outside the straight line with points on the straight line, the vertical line segment is the shortest.
7. The parallel axiom passes through a point outside the straight line, and there is only one straight line parallel to this straight line.
8. If two straight lines are parallel to the third straight line, then the two straight lines are also parallel to each other [1]
9. The same angle is equal, and two straight lines are parallel.
10, internal dislocation angles are equal, and two straight lines are parallel.
1 1, the inner angles on the same side are complementary, and the two straight lines are parallel.
12, two straight lines are parallel and have the same angle.
13, two straight lines are parallel and the internal dislocation angles are equal.
14. Two straight lines are parallel and complementary.
15, the sum of two sides of a theorem triangle is greater than the third side.
16, the difference between two sides of the inference triangle is smaller than the third side.
17, the sum of the internal angles of the triangle and the theorem triangle is equal to 180?
18, it is inferred that the two acute angles of 1 right triangle are complementary.
19, Inference 2 An outer angle of a triangle is equal to the sum of two inner angles that are not adjacent to it.
20. Inference 3 The outer angle of a triangle is larger than any inner angle that is not adjacent to it.
2 1, the corresponding edge of congruent triangles is equal to the corresponding angle.
22. The edge axiom (SAS) has two edges, and their included angle corresponds to the congruence of two triangles.
23. The corner axiom (ASA) has two corners and two triangles with equal corresponding sides.
24. Inference (AAS) has two angles, and the opposite side of one angle corresponds to the congruence of two triangles.
25, the edge axiom (SSS) has three edges corresponding to the equality and congruence of two triangles [2]
26. Axiom of hypotenuse and right-angled side (HL) Two right-angled triangles with hypotenuse and a right-angled side are congruent.
27. Theorem 1 The distance from the point on the bisector of the angle to both sides of the angle is equal.
28. Theorem 2 The point where two sides of an angle are equidistant is on the bisector of this angle.
29. The bisector of an angle is the set of all points with equal distance to both sides of the angle.
30, the nature theorem of isosceles triangle The two bottom angles of an isosceles triangle are equal (that is, equilateral angles)
3 1, inference 1 The bisector of the vertex of the isosceles triangle bisects the base and is perpendicular to the base.
32. The bisector of the top angle, the median line on the bottom edge and the height on the bottom edge of the isosceles triangle coincide with each other.
33. Inference 3 All angles of an equilateral triangle are equal, and each angle is equal to 60?
34. Decision theorem of isosceles triangle If a triangle has two equal angles, then the sides of the two angles are also equal (equal angles and equal sides).
35. Inference 1 A triangle with three equal angles is an equilateral triangle.
36. Inference 2 has an angle equal to 60? An isosceles triangle is an equilateral triangle.
37. In a right triangle, if an acute angle equals 30? Then the right angle it faces is equal to half of the hypotenuse.
38. The midline of the hypotenuse of a right triangle is equal to half of the hypotenuse.
39. Theorem The point on the vertical line of a line segment is equal to the distance between the two endpoints of this line segment.
40. The inverse theorem and the equidistant point between the two endpoints of a line segment are on the vertical line of this line segment.
4 1, the middle vertical line of a line segment can be regarded as the set of all points with equal distance at both ends of the line segment.
42. Theorem 1 Two graphs symmetric about a straight line are conformal.
43. Theorem 2 If two figures are symmetrical about a straight line, then the symmetry axis is the perpendicular line connecting the corresponding points.
44. Theorem 3 Two figures are symmetrical about a straight line. If their corresponding line segments or extension lines intersect, then the intersection point is on the axis of symmetry.
45. Inverse Theorem If the straight line connecting the corresponding points of two graphs is vertically bisected by the same straight line, then the two graphs are symmetrical about this straight line.
46. Pythagorean Theorem The sum of the squares of two right angles A and B of a right triangle is equal to the square of the hypotenuse C, that is, A 2+B 2 = C 2.
47. Inverse Theorem of Pythagorean Theorem If the three sides of a triangle are related to A 2+B 2 = C 2, then the triangle is a right triangle.
48. The sum of the internal angles of a quadrilateral is equal to 360?
49. The sum of the external angles of the quadrilateral is equal to 360?
50. Theorem The sum of the interior angles of a polygon is equal to (n-2)? 180?
The third summary of mathematical formula method in the second volume of the second day of junior high school
1. Pythagorean Theorem: The sum of squares of a right triangle and two right angles of A and B is equal to the square of hypotenuse C, that is, a2+b2=c2.
2. Inverse Pythagorean Theorem: If three sides of a triangle have a relationship a2+b2=c2, then the triangle is a right triangle.
3. Theorem: The sum of the internal angles of a quadrilateral is equal to 360? .
4. The sum of the external angles of the quadrilateral is equal to 360? .
5. Theorem The sum of the interior angles of a polygon is equal to (n-2)? 180? .
6. Inference: The external angle of any polygon is equal to 360? .
7. parallelogram property theorem 1: parallelogram pairs are equal.
8. parallelogram property theorem 2: the opposite sides of parallelogram are equal.
9. Inference: The parallel segments sandwiched between two parallel lines are equal.
10. parallelogram property theorem 1: diagonal bisection of parallelogram.
1 1. parallelogram property theorem 2: two groups of parallelograms with equal diagonals are parallelograms.
12. parallelogram property theorem 3: two groups of quadrangles with equal opposite sides are parallelograms.
13. parallelogram property theorem 4: A quadrilateral whose diagonal is bisected is a parallelogram.
14. parallelogram property theorem 5: A set of quadrilaterals with parallel and equal opposite sides is a parallelogram or quadrilateral.
15. rectangle property theorem 1: all four corners of a rectangle are right angles.
16. Rectangle property theorem 2: The diagonals of rectangles are equal.
17. Rectangle judgment theorem 1: A quadrilateral with a triangle forming a right angle is a rectangle.
18. Rectangle Decision Theorem 2: Parallelograms with equal diagonals are rectangles.
19. Diamond property theorem 1: All four sides of a diamond are equal.
20. Diamond Property Theorem 2: Diagonal lines of the diamond are perpendicular to each other, and each diagonal line bisects a set of diagonal lines.
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