Algebraic expression: an expression obtained by finite algebraic operations such as addition, subtraction, multiplication, division, multiplication and square root of numbers and letters representing numbers, or a mathematical expression containing letters is called algebraic expression. For example: ax+2b, -2/3, b 2/26, √a+√2, etc. Excluding equal sign (=, ≡), unequal sign (≡, ≤, ≥,,), approximate equal sign.
There can be an absolute value. For example: |x|, |-2.25|, etc.
Algebraic expressions are rational expressions without division or fraction, and rational expressions with division and fraction but without variables in division or denominator. A rational expression with letters in the denominator is called a fraction.
Algebraic expressions contain no roots and the denominator contains letters.
Addition and subtraction of algebraic expressions includes merging similar terms; Multiplication and division include basic operations, rules and formulas; Basic operations can be divided into power operation attributes; Laws can be divided into multiplication and division; Formulas can be divided into multiplication formula, zero exponential power formula and negative integer exponential correction formula.
Monomial and polynomial are collectively called algebraic expressions. For example, 2x/3 is a monomial. 0.4X+3 is a polynomial. X/y is not an algebraic expression, but a fraction.
The degree of a single high-order term is called the degree of a polynomial. Polynomials can be arranged in descending and ascending order.
Single index: refers to the sum of the times of each unknown in a single item. For example, the exponent of ab^3a^2, A is 1 time, B is 3 times, and C is 2 times, that is, 1+3+2=6 times, and the exponent is 6.
monomial
The concept of (1) single item
Algebraic expressions consisting of numbers multiplied by letters or letters multiplied by letters are called monomials. A single number or letter is also called a monomial, such as q,-1, a.
⑵ Single coefficient
1, and the constant factor in a single item is called the coefficient of this item.
2. If a single item contains only letter factors, the positive single item coefficient is 1 and the negative single item coefficient is-1.
(3) the number of monomials
1, the sum of all letter indices in a monomial is called the number of times of this monomial.
For example, the coefficient of 4xy is 4 and the degree is 2. The exponent of x is 1, the exponent of y is 1, and the exponents add up to 2.
Polynomial (1) Polynomial and Related Concepts
The sum of several monomials is called polynomial. In polynomials, each monomial is called a polynomial term, and the term without letters is called a constant term. Polynomials with several terms are called polynomial terms. Symbols in polynomials are considered as natural symbols of terms. A univariate polynomial of degree n has at most N+ 1 terms.
Example: In the polynomial 2x-3, 2x and -3 are his terms, where -3 is a constant; In polynomial x? Its term in +2x+ 18 is X &；; Sup2, 2x and 18, where 18 is a constant term.
⑵ Degree of polynomial
In a polynomial, the degree of the term with the highest degree is the degree of the polynomial.
(3) Polynomial arrangement
1. Sorting polynomials in descending alphabetical order is called sorting polynomials in descending alphabetical order. 2. Arranging a polynomial according to the exponent of a letter from small to large is called arranging polynomials according to the ascending power of this letter.
Since a polynomial is the sum of several monomials, we can use the addition algorithm to exchange the positions of the terms, while keeping the value of the original polynomial unchanged.
In order to facilitate the calculation of polynomials, a polynomial is usually arranged in a neat and simple form in a certain order, which is the arrangement of polynomials.
Pay attention to when doing polynomial arrangement problems:
(1) Since a single item includes its preceding attribute symbol, the attribute symbol of each item should still be regarded as a part of the item and moved together.
(2) The arrangement of polynomials with two or more letters should pay attention to:
A. first of all, it must be arranged according to the index of which letter.
B. determine whether to arrange letters inward or outward.
Addition and subtraction of algebraic expressions
Items with the same letters and times are called similar items.
When mastering the concept of similar items, we should pay attention to:
1. To judge whether several monomials or terms are similar, two conditions must be mastered:
(1) contains the same letters.
The same letter has the same number of times.
2. Similar items have nothing to do with coefficient or alphabetical order.
3. All constant terms are similar.
1. The concept of merging similar projects:
Merging similar terms in polynomials into one term is called merging similar terms.
2. Rules for merging similar projects:
The coefficients of similar items are added together, and the results are taken as coefficients, and the indexes of letters and letters remain unchanged.
3. To merge similar projects:
(1). Find out the similar items accurately.
⑵. The coefficients of similar items are added together (in brackets) by using the inverse distribution law, and the indexes of letters and letters remain unchanged.
(3). Write the result of the merger.
When mastering the merger of similar projects, we should pay attention to:
1. If the coefficients of two similar items are opposite, the result after merging similar items is 0.
2. Don't leave out items that can't be merged.
3. As long as there are no more similar terms, it is the result (either a monomial or a polynomial).
4. Merging several polynomials is not considered as merging similar terms [-3(a+b)c]+7(a+b)c=(7-3)(a+b)c, which is not called merging similar terms, but just using the method of merging similar terms.
5. The key to merging similar items is to correctly judge similar items. Example: 8a+2b+5a-b)
Solution: The original formula = (8+5) a+(2-1) b.
= 13a+b
13a+b； This "b" means1b. 1 and-1 are usually omitted, such as-1a= -a A.
Algebraic expression and multiplication of algebraic expression
Algebraic expressions can be divided into definitions and operations, definitions can be divided into monomials and polynomials, and operations can be divided into addition, subtraction, multiplication and division.
Addition and subtraction involve merging similar items. Multiplication and division include basic operations, rules and formulas. Basic operations can be divided into power operations. Rules can be divided into algebra and division, and formulas can be divided into multiplication formula, zero exponential power and negative integer exponential power.
Rule of power with base: multiply with base power and add with base index. A m× a n = A (m+n)。
Power law: power, constant basis, exponential multiplication. (a^m)^n=a^mn
The power law of product: the power of product is equal to the power obtained by multiplying each factor of product and then multiplying it. (AB) n = a n× b n。
The multiplication of monomials and monomials has the following rules: the monomials are multiplied by their coefficients and the same base respectively, and other letters and their exponents are kept as the factorial of the product.
There are the following rules for the multiplication of monomial and polynomial: the multiplication of monomial and polynomial means that each term of polynomial is multiplied by monomial, and then the products are added. a(m+n)=am+an。
Polynomial and polynomial multiplication have the following rules: polynomial and polynomial multiplication, first multiply each term of one polynomial with each term of another polynomial, and then add the obtained products. (a+b)(m+n)=am+an+bm+bn
The essence of polynomial monomial division is to convert polynomial monomial division into monomial division, so it is suggested to review and consolidate monomial division before learning this lesson.
The number of terms of the quotient obtained by dividing the polynomial by the monomial is the same as that of this polynomial, so don't miss the terms. To skillfully divide polynomials into monomials, we must master its basic operations. The operational nature of power is the basis of algebraic expression multiplication and division. Only by grasping this key step can we accurately divide polynomials into monomials. Symbol is still an important problem in operation. When dividing each term of a polynomial by a single term, we should pay attention to the symbols of each term and the symbols of each term.
Square difference formula: the product of the sum of two numbers and the difference between these two numbers is equal to the square difference between these two numbers. (a+b)(a-b)=(a-b)22 index
Complete square formula: the square of the sum of two numbers is equal to the sum of the squares of these two numbers, plus twice the product of these two numbers. The square of the difference between two numbers is equal to the sum of the squares of these two numbers, MINUS twice the product of these two numbers. (A B)2 = A 22ab+B 2；
Same base powers divides, the base remains the same, and the exponent is subtracted. a^m÷a^n=a^(m-n)
The zeroth power of any number not equal to zero is equal to 1. a^0= 1(a≠0)
The power of -p(p is a positive integer) of any number that is not equal to zero is equal to the reciprocal of the power of p of this number. A-p = 1/(a p) (a ≠ 0, p is a positive integer)
Key points of learning algebraic expressions
Algebraic formula is the most basic formula in algebra. It is necessary to introduce algebraic expressions and learn the following contents (such as fractions, quadratic equations with one variable, etc.). ). On the basis of studying rational number operation, simple algebraic expression, unary linear equations and inequalities, the algebraic expression is introduced. In fact, the relevant contents of algebraic expressions have been learned in the sixth grade, but now the contents of algebraic expressions are more applicable than in the past, which increases the background of practical application.
Block diagram of knowledge structure in this chapter:
There are many knowledge points in this chapter that are important or difficult. The key points and difficulties are as follows.
Four operations of algebraic expressions
Addition and subtraction of 1. algebraic expressions
Merging similar items is the key and difficult point. Not only is it difficult, but it is also often tested. When merging similar items, we should pay attention to the following three points: ① Only by mastering the concept of similar items can we distinguish similar items and accurately grasp the two standard letters and letter indexes for judging similar items;
(2) The definition of merging similar terms means merging similar terms in polynomials into one term. After merging similar terms, the number of terms in the formula will be reduced to achieve the purpose of polynomial;
(3) "Merging" refers to adding the coefficients of similar items, and the obtained results are used as new coefficients, and the letters and letter indexes of similar items should remain unchanged.
2. Multiplication and division of algebraic expressions
The emphasis is on multiplication and division of algebraic expressions, especially multiplication formulas. It is difficult for students to master the structural characteristics of multiplication formula and the broad meaning of letters in the formula. Therefore, it is difficult to use the multiplication formula flexibly, and the handling of symbols in brackets is another difficulty when adding brackets (or removing brackets). Parentheses (or brackets) are the deformation of polynomials, which should be carried out according to the rules of parenthesis (or brackets). In the multiplication and division of algebraic expressions, the multiplication and division of monomial is the key, because the multiplication and division of general polynomials should be "transformed" into the multiplication and division of monomial.
The main problems of the four operations of algebraic expressions are:
Four operations of (1) single item
This kind of questions mostly appear in the form of multiple-choice questions and application questions, which are characterized by examining four operations of a single item.
⑵ Operation of monomial sum polynomial
This kind of problems mostly appear in the form of solving problems, which are highly skilled and characterized by examining the four operations of monomials and polynomials.
Open the package and add the bracket.
There is a "+"before the brackets. Remove the brackets and the preceding "+",and nothing in the brackets will change the symbol.
Such as: b+(a+b)=b+a+b
There is a "-"before the brackets. Remove the brackets and the "-"in front, and change the symbols of everything in brackets.
Such as: a-(a-b)=a-a+b