Summary of Mathematics Knowledge Points of Grade One in Beijing Normal University
Polynomial divided by monomial
First of all, the monomial
1, an algebraic expression of the product of numbers and letters, is called a monomial.
2. The single numerical factor is called the single coefficient.
3. The index of all the letters in the monomial and the number of times called the monomial.
4. A single number or letter is also a monomial.
5. The coefficient of monomial with only letter factor is 1 or-1.
6. A number is a monomial, and its coefficient is itself.
7. The degree of a single nonzero constant is 0.
8. A single item can only contain multiplication or power operation, and cannot contain other operations such as addition and subtraction.
9. The coefficient of the monomial includes the symbol before it.
10, when the coefficient of the monomial is a fraction, it should be turned into a false fraction.
When 1 1 and the single coefficient is 1 or-1, the number "1" is usually omitted.
12, the number of monomials is only related to letters, and has nothing to do with the coefficient of monomials.
Second, polynomials
The sum of 1 and several monomials is called a polynomial.
2. Each monomial in a polynomial is called a polynomial term.
3. The term without letters in polynomial is called constant term.
4. A polynomial has several terms, which are called polynomials.
5. Every term of polynomial includes the symbol before the term.
6. Polynomials have no concept of coefficient, but have the concept of degree.
7. The degree of the degree term in a polynomial is called the degree of the polynomial.
Third, algebraic expressions.
1, monomials and polynomials are collectively called algebraic expressions.
2. Both monomials and polynomials are algebraic expressions.
3. Algebraic expressions are not necessarily monomials.
4. Algebraic expressions are not necessarily polynomials.
5. Algebraic expressions with letters in denominator are not algebraic expressions; It is a fraction to learn in the future.
Fourth, the addition and subtraction of algebraic expressions.
The theoretical basis of 1. Algebraic addition and subtraction is: the rule of removing brackets, the rule of merging similar items, and the multiplication distribution rate.
2. The key to the addition and subtraction of several algebraic expressions is to use the rule of brackets correctly, and then merge similar items accurately.
3, several general steps of algebraic expression addition and subtraction:
(1) List algebraic expressions: enclose each algebraic expression in parentheses and then connect it with a plus sign and a minus sign.
(2) Open brackets according to the rules for opening brackets.
(3) Merge similar items.
4, the general steps of algebraic evaluation:
(1) algebraic simplification.
(2) Substitution calculation
(3) For some special algebraic expressions, "whole substitution" can be used for calculation.
V. Multiplication with the same base number
1, multiplied by n identical factors (or factors) A, is recorded as an, and read as the n power (power) of A, where A is the base, n is the exponent, and the result of an is called the power.
2. Powers with the same base are called same base powers.
3. Same base multiplication algorithm: same base multiplication, constant base, exponential addition. Namely: am ﹒ an = am+n
This rule can also be reversed, that is, am+n = am-an.
5. Start the power of different cardinality. If it can be converted into a power with the same base, first turn it into a power with the same base, and then apply the rules.
Sixth, the power of power.
The sum of powers of 1 refers to the multiplication of several identical powers. (am)n represents the multiplication of n am.
2. Power algorithm: power, constant basis, exponential multiplication. (am)n=amn .
3. This rule can also be reversed, that is, AMN = (am) n = (an) m.
Seven, the product of power
1, the power of product is the power of cardinal number and product.
2. Multiplication algorithm of product: Multiplication of product is equal to multiplying each factor in the product separately, and then multiplying the obtained power. That is, (ab)n=anbn.
3. This rule can also be reversed, namely: AnBN = (ab) n.
Eight, the similarities and differences of three kinds of "power arithmetic"
1, * * * Similarities:
The cardinality in the (1) rule remains unchanged, and only the exponent is operated.
(2) The base (nonzero) and exponent in the law are universal, that is, they can be numbers or formulas (monomials or polynomials).
(3) For operations with three or more operations, the rule still holds.
2. Differences:
(1) The same base power multiplication is exponential addition.
(2) The power of the power is exponential multiplication.
(3) The product is multiplied by each factor and then multiplied by the result.
Nine, the same base power division
1, same base powers's division rule: same base powers division, base constant, exponential subtraction, namely: am÷an=am-n(a≠0).
2. This rule can also be reversed, that is, am-n=am÷an(a≠0).
Ten, zero exponential power
1, the meaning of zero exponential power: the power of 0 of any number not equal to 0 is equal to 1, that is, a0= 1(a≠0).
XI。 Negative exponential power
1, the -p power of any number that is not equal to zero is equal to the reciprocal of the p power of this number, that is:
Note: In same base powers's division, the base of zero exponential power and negative exponential power is not 0.
XII. Multiplication of Algebraic Expressions
(1) Multiplies the monomial by the monomial.
1, the rule of monomial multiplication: the monomial is multiplied by the monomial, and their coefficients are multiplied by the power of the same letter, respectively, and the remaining letters, together with their exponents, remain unchanged as the factors of the product.
2, coefficient multiplication, pay attention to symbols.
3. The powers of the same letters are multiplied, the base is unchanged, and the exponents are added.
4. For the letters only contained in the monomial, write them together with its index as the factor of the product.
5. The result of multiplying the monomial by the monomial is still the monomial.
6. The multiplication rule of monomials also applies to the multiplication of three or more monomials.
(2) Multiplication of monomial and polynomial
1. Multiplication rule of monomial and polynomial: Multiplying monomial and polynomial means multiplying each term in polynomial by monomial according to distribution rate, and then adding the products. Namely: m(a+b+c)=ma+mb+mc.
2. Please pay attention to the product logo when operating. Every term of a polynomial is preceded by a symbol.
3. The product is a polynomial with the same number of terms as the polynomial.
4. When mixing operations, pay attention to the operation sequence. If there are similar items in the results, they should be merged to get the simplest result.
(3) Multiplication of Polynomials and Polynomials
1, polynomial and polynomial multiplication rule: polynomial multiplication, first multiply each term of one polynomial with each term of another polynomial, and then add the products. That is: (m+n)(a+b)=ma+mb+na+nb.
2. Polynomial multiplication must not be repeated or missed. Multiplication should be carried out in a certain order, that is, every term of one polynomial should be multiplied by every term of another polynomial. Before merging similar terms, the number of terms of the product is equal to the product of two polynomial terms.
3. Every term of a polynomial is preceded by a symbol. When determining the symbols of terms in products, "the same number is positive and the different number is negative" should be applied.
4. If there are similar items in the operation results, they should be merged.
5. For multiplying two linear binomials with the coefficient of the linear term 1 containing the same letter, the following formula can be used to simplify the operation: (x+a)(x+b)=x2+(a+b)x+ab.
Thirteen, the square difference formula
1, (a+b)(a-b)=a2-b2, that is, the product of the sum of two numbers and the difference between these two numbers is equal to their square difference.
2. A and B in the square difference formula can be monomials or polynomials.
3. The square difference formula can be reversed, that is, a2-b2=(a+b)(a-b).
4. The square difference formula can also simplify the operation of the product of two numbers. To solve this kind of problem, we must first see whether two numbers can be converted into
(a+b)？ (a-b), and then see if a2 and b2 are easy to calculate.
Summary of Mathematics Knowledge Points of Grade One in Beijing Normal University
First, multiplication with the same base.
(m, n is an integer) is the most basic rule in power operation. When applying regular operations, the following points should be noted:
A) The prerequisite for using this rule is that when the bases of powers are the same and multiplied, the base a can be a specific number, letter, monomial or polynomial;
B) When the index is 1, don't mistake it for no index;
C) Don't confuse multiplication with addition of algebraic expressions. Multiplication, as long as the base is the same, the indexes can be added; For addition, not only the radix is the same, but also the exponent needs to be added;
Second, the power of power and the power of products.
Third, the division of power with the same base.
(1) The premise of applying the rule is that the cardinality is the same, and this rule can only be used if the cardinality is the same.
(2) Cardinality can be a specific number, or a monomial or polynomial.
(3) Exponential subtraction refers to subtracting the exponent of the divisor from the exponent of the divisor, and the difference is not negative.
Fourth, multiplication of algebraic expressions.
1, the concept of monomial: the algebraic expression composed of the product of numbers and letters is called monomial. A single number or letter is also a monomial. The numerical factor of a single item is called the coefficient of a single item, and the sum of all letter indexes is called the number of times of a single item.
For example, the coefficient of bca22- is 2-, the degree is 4, and the degree of a single nonzero number is 0.
2. Polynomial: The sum of several monomials is called polynomial. Each monomial in a polynomial is called a polynomial term, and the degree of the degree term is called the degree of the polynomial.
Five, the square difference formula
Expression: (a+b) (a-b) = a 2-b 2. The product of the sum of two numbers and the difference of two numbers is equal to the square of the difference of two numbers. This formula is called the square difference formula of multiplication.
Formula application
Can be used for some fractions whose denominator contains the root sign:
1/(3-4 root number 2) Simplification:
Six, the complete square formula
Common mistakes in the complete square formula are:
(1) missed a semester.
② Confusion formula
③ Symbol error in the operation result.
④ Variant application is difficult to master.
VII. Division of algebraic expressions
1, the division rule of monomial
In monomial division, the coefficient and the power of the same base number are separated as a factor of the quotient, and the letter only contained in the division formula, together with its exponent, is taken as a factor of the quotient.
Note: first determine the coefficient of the result (that is, coefficient division), and then divide it by the same base power. If only the letters in the division formula are included, it will be used as the factor of quotient together with its exponent.
Summary of Mathematics Knowledge Points of Grade One in Beijing Normal University
1. 1 positive and negative numbers
A number preceded by a minus sign "-"is called a negative number.
It has the opposite meaning to negative number, that is, I learned that numbers other than 0 are called positive numbers (sometimes "+"is added before positive numbers as needed).
1.2 rational number
Positive integers, 0 and negative integers are collectively called integers, and positive and negative fractions are collectively called fractions.
Integers and fractions are collectively called rationalnumber.
Numbers are usually represented by points on a straight line, which is called the number axis.
Three elements of number axis: origin, positive direction and unit length.
Take any point on a straight line to represent the number 0, and this point is called the origin.
Numbers with only two different signs are called opposites. (Example: the reciprocal of 2 is-2; The reciprocal of 0 is 0)
The distance between the point representing the number A on the number axis and the origin is called the absolute value of the number A, and it is recorded as |a|.
The absolute value of a positive number is itself; The absolute value of a negative number is its reciprocal; The absolute value of 0 is 0. Two negative numbers, the larger one has the smaller absolute value.
Addition and subtraction of rational number 1.3
Rational number addition rule:
1. Add two numbers with the same sign, take the same sign, and then add the absolute values.
2. Add two different symbols with different absolute values, take the symbol of the addend with larger absolute value, and subtract the one with smaller absolute value from the one with larger absolute value. Two opposite numbers add up to 0.
When a number is added with 0, it still gets this number.
Rule of rational number subtraction: subtracting a number is equal to adding the reciprocal of this number.
Multiplication and division of rational number 1.4
Rational number multiplication rule: two numbers are multiplied, the same sign is positive, the different sign is negative, and the absolute value is multiplied. Any number multiplied by 0 is 0.
Two numbers whose product is 1 are reciprocal.
Rational number division rule: dividing by a number that is not equal to 0 is equal to multiplying the reciprocal of this number.
Divide two numbers, the same sign is positive, the different sign is negative, and divide by the absolute value. Divide 0 by any number that is not equal to 0 to get 0. mì
The operation of finding the product of n identical factors is called power, and the result of power is called power. In the n power of a, a is called radix and n is called exponent.
The odd power of a negative number is negative and the even power of a negative number is positive. Any power of a positive number is a positive number, and any power of 0 is 0.
Scientific counting method is used to express numbers greater than 10 as the n power of a× 10.
From the first non-zero digit to the last digit on the left of a number, all digits are valid digits of this number.
Summary of first-year mathematics knowledge points in Beijing Normal University;
★ Review and summary of mathematics knowledge points in the second volume of the first day of Beijing Normal University Edition
★ Summary of knowledge points in the first volume of seventh grade mathematics in Beijing Normal University
★ Summary of Mathematics Knowledge in Junior Middle School of Beijing Normal University
★ Beijing Normal University Edition seventh grade mathematics knowledge points
★ Knowledge points in the first volume of first grade mathematics
★ Summary of Mathematics Knowledge Points below Grade 7 in Junior Middle School of Beijing Normal University
★ Seventh grade mathematics knowledge points Beijing Normal University Edition Volume 1
★ Guidance and summary of mathematics learning methods in senior one.
★ Summary of knowledge points of addition and subtraction of algebraic expressions
★ Beijing Normal University Junior High School Mathematics Knowledge Point Volume II