The related concepts and properties of key algebraic expressions, and the operation of algebraic expressions.

☆ Summary ☆

I. Key concepts

Classification:

1。 Algebraic and rational expressions

Formulas that associate numbers or letters representing numbers with operational symbols are called algebraic expressions. independent

Numbers or letters are also algebraic.

Algebraic expressions and fractions are collectively called rational forms.

2。 Algebraic expressions and fractions

Algebraic expressions involving addition, subtraction, multiplication, division and multiplication are called rational expressions.

Rational expressions without division or division but without letters are called algebraic expressions.

Rational number formula has division, and there are letters in division, which is called fraction.

3。 Monomial and polynomial

Algebraic expressions without addition and subtraction are called monomials. (product of numbers and letters-including single numbers or letters)

The sum of several monomials is called polynomial.

Note: ① According to whether there are letters in the division formula, algebraic expressions and fractions are distinguished; According to whether there are addition and subtraction operations in algebraic expressions, monomial and polynomial can be distinguished. ② When classifying algebraic expressions, the given algebraic expressions are taken as the object, not the deformed algebraic expressions. When we divide the category of algebra, we start from the representation. For example,

=x, =│x│ and so on.

4。 Coefficient and index

Difference and connection: ① from the position; (2) In the sense of representation.

5。 Similar projects and their combinations

Conditions: ① The letters are the same; ② The indexes of the same letters are the same.

Basis of merger: law of multiplication and distribution

6。 radical expression

The algebraic expression of square root is called radical.

Algebraic expressions that involve square root operations on letters are called irrational expressions.

Note: ① Judging from the appearance; ② Difference: It is a radical, but it is not an irrational number (it is an irrational number).

7。 arithmetic square root

(1) The positive square root of a positive number ([the difference between a ≥ 0-and "square root"]);

⑵ Arithmetic square root and absolute value

① Contact: all are non-negative, =│a│.

② Difference: │a│, where A is all real numbers; Where a is a non-negative number.

8。 Similar quadratic roots, simplest quadratic roots and denominator of rational numbers

After being transformed into the simplest quadratic root, the quadratic roots with the same number of roots are called similar quadratic roots.

The following conditions are satisfied: ① the factor of the root sign is an integer and the factor is an algebraic expression; (2) The number of roots does not include exhausted factors or factors.

Crossing out the root sign in the denominator is called denominator rationalization.

9。 index

(1)(- power supply, power supply operation)

(1) when a > 0, > 0; ② when a < 0, > 0 (n is even) and < 0 (n is odd)

(2) Zero index: = 1(a≠0)

Negative integer index: = 1/ (a≠0, p is a positive integer)

Second, the law of operation and the law of nature

1。 Rules of addition, subtraction, multiplication, division, multiplication and root of fractions

2。 Properties of fractions

(1) Basic properties: = (m≠0)

(2) Symbolic law:

⑶ Complex fraction: ① Definition; ② Simplified methods (two kinds)

3。 Algebraic expression algorithm (rules of removing brackets and adding brackets)

4。 The operational nature of power: ① =; ② ÷ = ; ③ = ; ④ = ; ⑤

Skills:

5。 Multiplication rule: (1) single× single; (2) single × many; 3 more x more.

6。 Multiplication formula: (plus or minus)

(a+b)(a-b)= 1

(a b) =

7。 Division rule: (1) single-single; (2) Too many orders.

8。 Factorization: (1) definition; ⑵ Methods: A. Common factor method; B. formula method; C. cross multiplication; D. group decomposition method; E. root formula method.

9。 Properties of arithmetic roots: =; ; (a≥0，b≥0)； (a ≥ 0, b > 0) (positive and negative)

10。 Radical operation rule: ⑴ addition rule (combining similar quadratic roots); (2) multiplication and division; (3) The denominator is reasonable: a; b； c .

1 1。 Scientific notation: (1 ≤ A < 10, n is an integer =

Third, the application examples (omitted)

Four, comprehensive operands (omitted)