1, the basic concept.
(1) binomial expansion: The polynomial on the right side of the equation is called (a+ b) "binomial expansion.
② Binomial coefficient: the coefficient of each term in the c% (C%(r = 0, 1, 2, ... n n) expansion.
③ Number of terms: the expansion term r+ 1 is a homogeneous polynomial about A and B. ..
④ General term: the r+ 1 term of the expansion is marked as tr+1= c% an-Rb "(r = 0.1.2 ... n).
2. A few reminders.
① Number of items: the extension * * * has n+ 1 item.
② Order: Pay attention to the correct choice of A and B, and the order cannot be changed, that is, (a+b)n and (b+a)n are different.
③ Exponent: The exponent of A ranges from n to 0. Descending order; The exponent of b is arranged from 0 to n in ascending order. The sum of the indices of a and b in each term is always n.
④ Coefficient: correctly distinguish binomial coefficient from binomial coefficient: binomial coefficient refers to the number of combinations before each term; The coefficient of a term refers to the part of each term that excludes variables (including binomial coefficient).
Brief introduction of binomial theorem;
The binomial theorem (Newton's binomial theorem) was put forward by isaac newton in 1664 and 1665. The binomial theorem points out that the integer power of the sum of two numbers, such as the identity expanded into the sum of similar terms, can be extended to any real power.
The binomial theorem was originally used to open higher powers. In China, Nine Chapters Arithmetic, written in 1 century AD, put forward the world's earliest general program for finding the square root and square root of multiple positive integers. 165438+In the middle of the 20th century, Jia Xian gave the original drawing of "root-tapping method" in his book "Unlocking Calculation", which met the need of root-tapping for more than three times.
This graph is a table of binomial coefficients of the sixth power. However, Jia Xian did not give the general formula of binomial coefficient, so he failed to establish the binomial theorem of general positive integer power. In the 3rd century A.D./KLOC-0, Yang Hui quoted this number in his Nine Chapters Detailed Explanation of Algorithms, and indicated that this number came from Jia Xian's Unlocking Calculation Book.
Jia Xian's works have been lost, but Yang Hui's works have survived to this day, so this map is called "Jia Xian Triangle" or "Yang Hui Triangle". /kloc-At the beginning of the 4th century, Zhu Shijie reprinted this picture in his Four Yuan Jade Sword, and added two layers and two parallel diagonals.