A rational expression in algebraic expression. If there is no division or fraction, if there is a division and fraction, but there is no variable in the division or denominator, it is called an algebraic expression. If there is a division operation with letters, then the formula is called fractional decimal. )
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.
Algebraic expressions and similar items
1. Single item
The expression of (1) monomial: 1, the product of a number and a letter. This algebraic expression is called monomial 2, and a single letter is also a monomial.
3. A number is a monomial. 4. Letters multiplied by letters form a monomial. 5. Multiply numbers to form a monomial.
(2) Single coefficient: the numerical factors and property symbols in a single item are called single coefficient.
If a single item contains only numerical factors, the positive single item coefficient is 1 and the negative single item coefficient is-1.
(3) The number of monomials: The sum of the indices of all the letters in the monomials is called the number of monomials.
2.polynomial
The concept of (1) polynomial: 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 polynomials. The symbols in polynomials are regarded as the natural symbols of each term. A univariate polynomial of degree n can have at most N+ 1 terms.
(2) Degree of Polynomial: The degree of the term with the highest degree in the polynomial is the degree of the polynomial.
(3) the arrangement of polynomials:
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, the position of each term can be exchanged by the addition algorithm, 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 contains 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.
(3) Algebraic expression:
Monomial and polynomial are collectively called algebraic expressions.
(4) the concept of similar items:
Items with the same letters and times are called similar items, and several constant items are also 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. Several constant terms are similar.
(5) Merge similar items:
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 similar projects accurately.
(2) Reverse the distribution law, add the coefficients of similar items together (enclosed in brackets), and keep the letters and their indices unchanged.
(3) Write the merged result.
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 items, it is the result (either a single item or a polynomial).
The key to merging similar items: correctly judging similar items.
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.
The power rule of the same base: multiply with the power of the same base and add with the index of the same base.
Power law: power, constant basis, exponential multiplication.
Power law of product: the power of product is equal to the power obtained by multiplying the factors of product respectively and then multiplying them.
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 is to multiply each term of polynomial with monomial, and then add the products.
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.
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.
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.
Same base powers divides, the base remains the same, and the exponent is subtracted.
On the learning points of algebraic expressions
Tu Xinmin
Algebraic formula is the most basic formula in algebra, so it is necessary to introduce algebraic formula and learn the following contents (such as fractions, quadratic equations with one variable, etc.). ). On the basis of studying rational number operations, simple algebraic expressions, linear equations and inequalities, algebraic expressions are 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.
First, the four operations of algebraic expressions
Addition and subtraction of 1. algebraic expressions
Merging similar items is the key and difficult point. 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) It is clear that the meaning of merging similar terms is to merge similar terms in polynomials into one term. After merging similar terms, the number of terms in the formula will be reduced, thus simplifying the 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, the flexible application of multiplication formula is difficult, and the handling of symbols in brackets is another difficulty when adding (or removing) brackets. Parentheses (or brackets) are the deformation of polynomials, which should be carried out according to the law of parenthesis (or brackets). In the multiplication and division of algebraic expressions, the single multiplication and division is the key, because the multiplication and division of general polynomials should be "transformed" into the single multiplication and division.
The main problems of the four operations of algebraic expressions are:
Four operations of (1) monomial
This kind of questions mostly appear in the form of multiple-choice questions and application questions, which are characterized by examining four operations of monomials.
(2) Operation of monomial and 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.
Second, factorization.
The difficulty is the four basic methods of factorization (raising common factor, using formula, grouping factorization and cross multiplication). Factorization is the reverse deformation of algebraic expression multiplication, and the introduction of factorization should firmly grasp this point.