(1), understand and master the binomial theorem, and skillfully write the general term of binomial expansion, and use this general term to solve the problem of finding the specified term and the coefficient of the specified term, and correctly distinguish the concepts of binomial coefficient and binomial expansion term.
(2) Understand and master the mathematical idea of binomial theorem derivation, and use it to solve similar problems of polynomials (such as simplifying three terms into two terms), and be familiar with the application of binomial theorem in seeking approximation, proving divisibility and proving inequality.
(3) Requirements and trends of the college entrance examination: in the college entrance examination, it usually appears in the form of multiple-choice questions or fill-in-the-blank questions, mostly the application of general items and the nature and application of binomial coefficients; But now there are signs of infiltrating comprehensive series and function propositions into big questions.
Second, the basic knowledge system
① formula: (a+b) n =++………+(n ∈ n *)
②, I), general formula: Tr+ 1=Crn? Ann -r? Br is r+ 1, in descending order of A and ascending order of B.
Ⅱ) Pay attention to the difference between the binomial coefficient of the expansion and the coefficient of the item in the expansion.
Ⅲ) Common anomalies: (1+x) n =1++...+; ( 1-x)n= 1- + +…
Main methods to deal with problems: specific problems, such as constant terms, x2, etc. Deduct the general term; Coefficients and problems in expansion? Distribution method
③ The main properties of binomial coefficient:
(1), symmetry =
(2), increase or decrease and maximum: pay attention to the difference between the maximum binomial coefficient and the maximum expansion coefficient; When n is odd, the binomial coefficients of the middle two terms are equal, and the maximum value is obtained at the same time; When n is an even number, the binomial coefficient of the middle term takes the maximum value (binomial coefficient increases first and then decreases, taking the middle maximum value).
(3) binomial coefficient summation formula →++…+= 2n; In the expansion of (a+b)n, the sum of binomial coefficients of odd terms is equal to the sum of binomial coefficients of even terms; The formula is →++… =+… = 2n- 1.
(4), polynomial? What is the sum of the coefficients of (x)? ( 1); Polynomial? What is the sum of the coefficients of the odd terms of (x)? ( 1)-? (-1)2. Polynomial? What is the sum of the even coefficients of (x)? ( 1)+? (- 1)2; This is essentially the result of assignment.
(5) In binomial expansion, the method to find the term with the largest coefficient→ comparison method, that is, the coefficients are Pr, Pr+ 1, PR-1respectively; then what Pr maximum
Third, the analysis of common questions and the understanding of rules, methods and skills
(1) Use the general formula to solve the specific item problem in the expansion.
The basic method to find a term of binomial expansion or a term that meets certain conditions and has certain properties is to analyze and discuss the solution by using the general term formula of binomial.
★ title1(National Ⅰ in 2006? In the expansion of (x- 12x) 10, the coefficient of x4 is ().
a- 120 B 120 C- 15D 15
● The coefficient between the solution and x4 is C3 10(- )3 =- 15 ★ Problem 2 in the expansion of binomial (3x-2x)15, ① the constant term is _ _; ② How many rational terms are there? ______; (3) There are several kinds of algebraic expressions.
Low solution, ① the general term of expansion is; ② When r = 6,30-5R6 = 0, the constant term is T7 = 26C615; ; ③ When 30-5r6 = 5-56 r is an integer, then r can take three numbers, 0,6, 12, so * * * has three rational terms; ④ When 5-56 r is a non-negative integer, r = 0 or 6, so there are two integers.