I. Definition
1 algebraic expression multiplication
(1).am an = am+n [m, n are all positive integers]
Power with the same base, the base is constant, and the index is added.
(2).(am)n=amn[m, n are all positive integers]
Power, constant radix, exponential multiplication.
(3).(ab)n=anbn[n is a positive integer]
The power of a product is equal to multiplying the factors of the product, and then multiplying the power.
(4). ac5 bc2 =(a b)(C5 C2)= ABC 5+2 = ABC 7
Multiply the monomial with the monomial, respectively by their coefficients and the same letters. For letters contained only in the monomial, they are used as a factor of the product together with its index.
(5).m(a+b+c)=ma+mb+mc
Multiplying a polynomial by a monomial is to multiply each term of a polynomial by a monomial, and then add the products.
(6).(a+b)(m+n)=am+an+bm+bn
Multiply each term of one polynomial by each term of another polynomial, and then multiply the products.
2. Multiplication formula
( 1).(a+b)(a-b)=a2-b2
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.
(2).(a b)2=a2 2ab+b2
Complete square formula: the square of the sum [or difference] of two numbers is equal to the sum of their squares, plus [or minus] twice their product.
3. Algebraic expression division
(1) am ÷ an = am-n [a ≠ 0, m and n are positive integers, m >；; n]
Same base powers divides, the base remains the same, and the exponent is subtracted.
(2)a0= 1[a≠0]
Any number that is not equal to the power of 0 is equal to 1.
(3) Monomial division, in which the coefficient and the same base power are separated as a factor of the quotient, and only the letters contained in the division formula and their exponents are used as a factor of the quotient.
(4) Divide the polynomial by the monomial, first divide each term of the polynomial by the monomial, and then add the obtained quotients.
Decomposition of a polynomial into the product of several algebraic expressions is also called decomposition of this polynomial.
Two. main points
1.(x+p)(x+q)=x2+(p+q)x+pq
2.x3-y3=(x-y)(x2+xy+y2)
3. Two basic methods of factorization:
(1) Method for extracting common factors. Extraction: Numbers are the common divisor of entries, each entry contains letters, and the index is the lowest among the entries.
(2) Formula method.
(1) A2-B2 = (a+b) (a-b) The square difference of two numbers is equal to the product of the sum of these two numbers and the difference of these two numbers.
② A2 2ab+B2 = (A b) 2 The sum of squares of two numbers plus [or minus] twice the product of these two numbers is equal to the square of the sum [or difference] of these two numbers.