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.

1. Single item

(1) The concept of monomial: Algebraic expressions such as the product of numbers and letters are called monomials, and a single number or letter is also a monomial.

Note: Numbers and letters have a product relationship.

(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. decide whether to arrange according to this letter.

(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.

note:

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.