The second volume of mathematics in the second day of junior high school summarizes the knowledge points I. Factorization
1. Transforming 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 common factor method, such as ab+ac=a(b+c).
2. Concept connotation: (1) The final result of factorization 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, that is, ma+mb-mc=m(a+b-c).
3. Comments on error-prone points: (1) Note whether the sign and power exponent of the item are wrong; (2) Whether the common factor formula is "clean";
(3) One term in the polynomial is only a common factor. After it is put forward, this item in brackets is+1, and there is no leakage.
Three. Using 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:
4. Use the formula method:
(1) square difference formula: ① It should be binomial or polynomial regarded as binomial; (2) Every term of binomial (unsigned) is the square of monomial (or polynomial); Binomials are different symbols.
(2) Complete square formula: ① It should be a trinomial formula; (2) where two numbers are the same and each is the square of an algebraic expression;
③ Another term can be positive or negative, and it is twice the base product of the first two terms.
5. Factorization ideas and problem-solving steps:
(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 every factorization can no longer be decomposed within the scope of rational numbers.
Key knowledge of mathematics in the second day of junior high school
I. Parallelogram
Properties of (1) parallelogram
1) Definition of parallelogram: A parallelogram with two groups of parallel opposite sides is called a parallelogram.
2) Properties of parallelogram (including sides, angles and diagonals):
Side: ① Two groups of opposite sides of parallelogram are parallel respectively;
② Two groups of opposite sides of parallelogram are equal respectively;
Angle: ③ The two groups of diagonals of parallelogram are equal respectively;
Diagonal line: ④ The diagonal line of parallelogram is equally divided.
Complementary adjacent angles of parallelogram; A parallelogram is a figure with a symmetrical center, and the symmetrical center is the intersection of diagonals.
(2) parallelogram judgment
1) determination of parallelogram (including sides, angles and diagonals);
Side: ① Two groups of parallelograms with parallel opposite sides are parallelograms;
② Two groups of quadrangles with equal opposite sides are parallelograms;
(3) A group of parallelograms with equal opposite sides are parallelograms;
Angle: ④ Two groups of quadrangles with equal diagonals are parallelograms;
Diagonal lines: ⑤ A quadrilateral with diagonal lines bisecting each other is a parallelogram.
2) triangle midline: the line segment connecting the midpoints of the two sides of the triangle is called the triangle midline.
3) Triangle midline theorem: the midline of a triangle is parallel to the third side of the triangle and equal to half of the third side.
4) Distance between parallel lines:
In two parallel lines, the distance between any point on a straight line and another straight line is called the distance between these two parallel lines. The distance between two parallel lines is equal everywhere.
Ⅱ. Rectangular shape
Properties of (1) rectangle
Definition of rectangle: A parallelogram with a right angle is called a rectangle.
2) the nature of the rectangle:
① A rectangle has all the properties of a parallelogram;
② All four corners of a rectangle are right angles;
③ The diagonals of the rectangles are equal;
(4) The rectangle is both an axisymmetric figure and a centrally symmetric figure, with two symmetrical axes, and the symmetrical center is the intersection of diagonal lines.
(2) Determination of rectangle
1) rectangle determination:
① A parallelogram with a right angle is a rectangle;
② Parallelograms with equal diagonals are rectangles;
A quadrilateral with three right angles is a rectangle.
2) Steps to prove that a quadrilateral is a rectangle:
Method 1: First, prove that a quadrilateral is a parallelogram, and then prove that an angle is a right angle or diagonal is equal;
Method 2: If a quadrilateral has multiple' right angles', it can be proved that three angles are right angles.
3) Right triangle hypotenuse midline theorem: (as shown on the right)
The center line of the hypotenuse of a right triangle is equal to half of the hypotenuse.
Ⅲ. Diamond shape
Properties of (1) rhombus
1) Definition of rhombus: A group of parallelograms with equal adjacent sides is called a rhombus.
2) the nature of the diamond:
① A diamond has all the properties of a parallelogram;
② All four sides of the diamond are equal;
③ The two diagonals of the diamond are perpendicular to each other, and each diagonal bisects a set of diagonals;
(4) The rhombus is both an axisymmetric figure and a centrally symmetric figure, with two symmetrical axes, and the symmetrical center is the diagonal intersection.
3) area formula of diamond:
Then, the lengths of the two diagonal lines of the diamond are respectively
(2) Determination of diamond shape
1) diamond judgment:
① A set of parallelograms with equal adjacent sides is a rhombus;
② Parallelograms with mutually perpendicular diagonals are rhombic;
A quadrilateral with four equilateral sides is a diamond.
2) the steps to prove that the quadrilateral is a diamond:
Method 1: First prove that it is a parallelogram, and then prove that "a group of adjacent sides are equal" or "diagonals are perpendicular to each other";
Method 2: Directly prove "four sides are equal".
ⅳ. Square
The Properties of (1) Square
1) Definition of a square: A group of parallelograms with equal adjacent sides and a right angle is called a square.
2) the nature of the square:
A square has all the properties of parallelogram, rectangle and diamond, that is, ① all four sides of a square are equal; ② All four corners are right angles; (3) Diagonal lines are vertically bisected and equal, and each diagonal line bisects a set of diagonal lines.
3) A square is not only an axisymmetric figure, but also a centrally symmetric figure. It has four axes of symmetry, and the intersection of diagonal lines is the center of symmetry.
(2) the judgment of the square
1) square of judgment:
① A group of parallelograms with equal adjacent sides and a right angle is a square;
② A group of rectangles with equal adjacent sides are squares;
③ Rectangles with diagonal lines perpendicular to each other are squares;
④ A diamond with a right angle is a square;
⑤ The rhombus with equal diagonal lines is a square;
⑥ A quadrilateral whose diagonal is vertically bisected is a square.
Mathematics knowledge in the second day of junior high school is often tested.
1. triangle: A figure composed of three line segments that are not on the same line and are connected end to end is called a triangle.
2. Trilateral relationship: the sum of any two sides of a triangle is greater than the third side, and the difference between any two sides is less than the third side.
3. Height: Draw a vertical line from the vertex of the triangle to the line where the opposite side is located, and the line segment between the vertex and the vertical foot is called the height of the triangle.
4. midline: in a triangle, the line segment connecting the vertex and the midpoint of its opposite side is called the midline of the triangle.
5. Angular bisector: The bisector of the inner angle of a triangle intersects the opposite side of this angle, and the line segment between the intersection of the vertex and this angle is called the angular bisector of the triangle.
6. Stability of triangle: The shape of triangle is fixed, and this property of triangle is called stability of triangle.
7. Polygon: On the plane, a figure composed of end-to-end line segments is called polygon.
8. Interior Angle of Polygon: The angle formed by two adjacent sides of a polygon is called its interior angle.
9. Exterior angle of polygon: The angle formed by the extension line of one side of polygon and its adjacent side is called the exterior angle of polygon.
10. Diagonal line of polygon: the line segment connecting two nonadjacent vertices of polygon is called diagonal line of polygon.
1 1. Regular polygon: A polygon with equal angles and sides in a plane is called a regular polygon.
12. plane mosaic: a part of a plane is completely covered by some non-overlapping polygons, which is called covering the plane with polygons.
13. Formulas and properties:
⑴ Sum of triangle internal angles: The sum of triangle internal angles is 180.
(2) the nature of the triangle exterior angle:
Attribute 1: One outer angle of a triangle is equal to the sum of two non-adjacent inner angles.
Property 2: The outer angle of a triangle is larger than any inner angle that is not adjacent to it.
⑶ Formula for the sum of polygon internal angles: the sum of polygon internal angles is equal to 180.
⑷ Sum of polygon external angles: The sum of polygon external angles is 360.
5] Diagonal number of polygons: ① Diagonal lines can be drawn from a vertex of a polygon.
Further reading: the knowledge point of the second volume of ninth grade mathematics 1, and the conditions for the establishment of quadratic roots: the number of roots is non-negative.
2. The essence of quadratic root: it is a non-negative arithmetic square root. Therefore √a≥0.
3. Two formulas: (√ a) 2 = a (a ≥ 0); √a2=∣a∣.
4. Quadratic radical multiplication and division: √a ×√b=√ab(a≥0, b ≥ 0); √a÷√b=√a/b(a≥0，b & gt0).
5. The simplest quadratic root formula: (1) The number of square roots does not contain denominator; ⑵ The number of square roots does not contain factors that can be opened.
6. Addition and subtraction of quadratic roots: first convert quadratic roots into the simplest quadratic roots, and then merge quadratic roots with the same number of roots.
7. Use the formula: (a+b) (a-b) = A2-B2; (a b)2=a2 2ab+b2。
Chapter 22 One-variable quadratic equation
1, definition: an equation in the form of ax2+bx+c=0(a≠0) is called a quadratic equation.
① is an integral equation, ② is quadratic with the highest number of unknowns, ③ contains only one unknown, and ④ the coefficient of quadratic term is not zero.
2. The general form of unary quadratic equation: descending order, the coefficient of quadratic term is usually positive, and the right end is zero.
3. The root of a quadratic equation: substitution makes the equation hold.
4. Solution of the unary quadratic equation: ① collocation method: shift the term → turn the quadratic term coefficient into one → add half of the linear term coefficient on both sides at the same time → formula → square root → write the solution of the equation.
② Formula: x = (-b √ B2-4ac)/2a. ③ Factorization method: the right end is zero, and the left end is decomposed into the product of two factors.
5. Discriminant formula of roots of quadratic equation in one variable: ① When △ >; 0, the equation has two unequal real roots, ② when △ = 0, the equation has two equal real roots, ③ when △
Note: the prerequisite for application is: a≠0.
6. The relationship between the root and the coefficient of a quadratic equation: x 1+x2 =-b/a, x1* x2 = c/a. 。
Note: the prerequisite for application is: a ≠ 0, delta ≥ 0.
7. Solving application problems with column equations: examining questions and setting elements → column algebra and column equations → arranging them into general forms → solving equations → testing and answering questions.
Chapter 23 Rotation
1. Three elements of rotation: rotation center, rotation direction and rotation angle.
2. The essence of rotation: ① The distance from the corresponding point to the center of rotation is equal; ② The included angle between the corresponding point and the connecting line of the rotation center is equal to the rotation angle; ③ The numbers before and after rotation are the same.
Key: Find the corresponding line segment and the corresponding angle.
3. Center symmetry: rotate the figure around a certain point 180. If it can coincide with another graph, then the two graphs are symmetric or central about this point.
4. The essence of central symmetry: ① Two figures with central symmetry, the connecting line segments of the corresponding points all pass through the symmetrical center and are equally divided by the symmetrical center. (2) congruence of two graphs with central symmetry.
5. Centrally symmetric figure: rotate the figure around a certain point 180. If the rotated figure can coincide with the original figure, then this figure is called a central symmetric figure.
6. The coordinate law of the symmetrical point: ① Axis symmetry about X: the abscissa is unchanged, but the ordinate is opposite; ② Axis symmetry about Y: the abscissa is opposite and the ordinate is unchanged; ③ Symmetry of origin: the abscissa and ordinate are opposite.
Chapter 24 Circle
1. Conditions for determining a circle: center → position, radius → size.
2. Concepts related to circle: chord diameter, arc semicircle, upper arc, lower arc, central angle, circumferential angle and chord center distance.
3. Symmetry of the circle: The circle is both an axisymmetric figure and a centrally symmetric figure.
4. Vertical diameter theorem: the diameter perpendicular to the chord bisects the chord and bisects the two arcs opposite the chord.
Inference: The diameter (not the diameter) of bisecting the chord is perpendicular to the chord and bisects the two arcs opposite the chord.
5. Relationship among central angle, arc, chord and chord center distance: In the same circle or within the same circle, the central angle of the circle has equal arc, chord and chord center distance.
Extension: among these four groups, as long as one group is equal, the other groups are equal.
6. Theorem of circumferential angle: ① The circumferential angle is equal to half of the central angle of the same arc,
(2) In the same circle or equal circle, the circumferential angle of the same arc or equal arc is equal, which is equal to half the central angle of the arc; Equal circumferential angles face equal arcs,
(3) The circumference angle subtended by the semicircle (or diameter) is a right angle, and the chord subtended by the circumference angle of 90 is the diameter.
7. Inner heart and outer heart: ① The inner heart is the intersection point of the bisector of the inner angle of the triangle, and its distance to the three sides of the triangle is equal.
(2) The epicenter is the intersection of the perpendicular lines of the three sides of the triangle, and its distance to the three vertices of the triangle is equal.
8. positional relationship between straight line and circle: intersection →d
9. Determination of tangent: "A small point is connected with the center of the circle" → Prove verticality. "Nothing is vertical" → Prove D = R.
The nature of the tangent: the tangent of the circle is perpendicular to the radius passing through the tangent point.
10, tangent length theorem: two tangents of a circle are drawn from a point outside the circle, and their tangent lengths are equal. The connecting line between this point and the center of the circle bisects the included angle of the two tangents.
1 1, the properties of inscribed quadrangles: the diagonals of inscribed quadrangles are complementary, and each outer angle is equal to its inner diagonal.
12, the properties of the circular circumscribed quadrilateral: the sum of the opposite sides of the circular circumscribed quadrilateral is equal.
13, the positional relationship between circles: outward → d > R+R. Circumtangent → D = R+R. Intersection → R-R-R.
14, regular polygon and circle: radius → circumscribed circle radius, central angle → central angle of each side, apothem → distance from center to one side.
15, arc length and sector area: L=n∏R/ 180. S sector =n∏R2/360.
16. lateral area and total area of the cone: generatrix length of the cone = radius of the sector, circumference of the bottom of the cone = arc length of the sector, lateral area of the cone = sector area, and total area of the cone = sector area+circular area of the bottom.
Chapter 25 Preliminary Probability
1, three kinds of events: random events, impossible events and inevitable events.
2. Probability: P(A)=p. 0≤P(A)≤ 1.
3. Solution of classical probability: ① enumeration method (indicating all possible results), ② list method, ③ tree diagram.
4. Estimate probability by frequency: According to the constant that the frequency of a random event is gradually stable, the probability of this event can be estimated.
Chapter 26 Quadratic Function
1. definition: a function in the form of y=ax2+bx+c(a≠0, a, b and c are constants) is called a quadratic function.
2. Classification of quadratic function: ①y=ax2: vertex coordinate: origin; Symmetry axis: y axis;
②y=ax2+c: vertex coordinates: (0, c); Symmetry axis: y axis;
③y=a(x-h)2: vertex coordinates: (h, 0); Symmetry axis: straight line x = h；;
④y=a(x-h)2+k: vertex coordinates: (h, k); Symmetry axis: straight line x = h；;
⑤y=ax2+bx+c: vertex coordinates: (-b/ 2a, 4ac-b2/4a); Symmetry axis: straight line x=-b/ 2a
3. Determination of symbols A, B and C: A: the opening direction is upward → A > 0; Opening direction downward → a
B: The left and right are different from A, the symmetry axis is on the left side of Y axis, and A and B have the same sign; The symmetry axis is on the right side of the Y axis, and the signs of A and B are different.
C: positive semi-axis intersecting with Y axis, C >；; 0; Negative semi-axis c intersecting with y axis
B2 -4ac: the number of times intersecting with the X axis, △ > 0→ two intersections, △
3. Translation methods: "positive left and negative right" and "positive upper and negative lower".
Premise: the formula is in the form of y = a (x-h) 2+k.
4. Determine the function relationship with undetermined coefficient method: ① Select y=ax2 as the vertex at the origin;
(2) Vertex Y = AX2+C on the Y axis;
③ Select y = ax2+bx； by coordinate origin;
④ Knowing that the vertex is selected on the X axis as y = a (x-h) 2;
⑤ Choose y = a (x-h) 2+k when the coordinates of vertices are known;
⑥ If you know the coordinates of three points, choose Y = AX2+BX+C.
5. Other applications: finding the intersection point with the X axis → solving a quadratic equation with one variable; The intersection with the y axis is (0, c).
6. Symmetry law: ① Two parabolas are symmetrical about X: A, B and C are their opposites.
② Two parabolas are symmetrical about Y axis: A and C remain unchanged, and B becomes its opposite number.
7. Practical problem: profit = sales volume-total purchase price-other expenses, profit = (selling price-purchase price) * sales volume-other expenses.