1. Cross multiplication concept Cross multiplication can factorize some quadratic trinomials. The key of this method is to decompose the quadratic coefficient A into the product of two factors a 1 and A2 A 1. 6? 1a2, decompose the constant term c into the product of two factors, c 1 and C2? 6? 1c2, and make a 1c2+a2c 1 just be a linear term b, then the result can be written directly: when factorizing factors in this way, we should pay attention to observation, try and realize that it is actually the inverse process of binomial multiplication. When the first coefficient is not 1, it often needs to be tested many times, so be sure to pay attention to the sign of each coefficient. Example 1 Factorization 2x 2-7x+3. Analysis: First, the quadratic term coefficient is decomposed and written in the upper left corner and the lower left corner of the crosshair, then the constant term is decomposed and written in the upper right corner and the lower right corner of the crosshair, and then the algebraic sum is obtained by cross multiplication to make it equal to the linear term coefficient. Decomposition of quadratic coefficient (positive factor only): 2 = Decomposition of constant term: 3 =/kloc-0 /× 3 = 3×1= (-3 )× (-1)× (-3) The following four cases are represented by crossing lines:/kloc-0. Kloc-0/ 3╳2 After observation, 8+0× (-3)+2× (-1) =-51-3 ╳ 2-1× (-/kloc-) The algebraic sum of the two terms is exactly equal to the coefficient of the first term -7. Solution 2x 2-7x+3 = (x-3) (2x- 1). Generally speaking, for the quadratic trinomial ax2+bx+c(a≠0), if the coefficient of the second term A can be decomposed into the product of two factors, that is, A = A2, c 1, c2, the arrangement is as follows: A1C1╳ AC2C2. Then add them to get a 1 C2+a2c 65438+ 1 a 1c2+a2c 1=b, then the quadratic trinomial can be decomposed into a 1x+c 1 and a2x+c2. It is often called cross multiplication. Example 2 decomposes 6x2-7x -5 into factors. Analysis: According to the method of example 1, the quadratic term coefficient 6 and the constant term -5 are decomposed and arranged respectively. There are eight different arrangements, one of which is 2 1 ╳ 3-5 2× (-5)+3. Therefore, the original polynomial can be factorized by cross multiplication. The solution 6x 2-7x-5 = (2x+ 1) (3x-5) points out that it often takes many observations to decompose a quadratic trinomial whose quadratic coefficient is not 1 by cross multiplication. To determine whether cross multiplication can be used for factorization. For the quadratic trinomial with quadratic coefficient of 1, cross multiplication can also be used to decompose the factors. At this time, you only need to consider how to decompose the constant term. For example, factorize X 2+2x- 15. The cross multiplication is1-3 ╳151× 5+1× (-3) = 2, so x 2+2x- 15 = (x-3) (x+5 When we decompose the coefficients of quadratic term and constant term, we only need to decompose 5 and -8. After using the crosshair to decompose, we can choose a suitable group through observation, that is,12 ╳ 5-41× (-4)+5× 2 = 6 to solve 5x 2+6xy-. Only multiply the polynomials first, and then factorize the deformed polynomials. Q: What are the characteristics of the factor of the product of two factors? What is the simplest method of polynomial multiplication? Answer: If we put forward the common factor 2 for the first two terms in the second factor, it will become 2(x-y), which is twice that of the first factor, and then multiply (x-y) as a whole, we can transform the original polynomial into a quadratic trinomial about (x-y). Cross multiplication can be used to decompose factors. Solution (x-y) (2x-2y-3)-2 = (x-y) [2 (x-y)-3]-2 = 2 (x-y) 2-3 (x-y)-2 = [(x-y)]. .1-2 ╳ 211×1+2× (-2) =-3 points out that decomposing the (x-y) factor into a whole is another application of the "whole" thinking method in mathematics. Example 5 x 0 can be decomposed into the product of two numbers with different signs, which can be decomposed into (-1)( 15), or (1)(- 15) or (3) (-5) or (-3)(5). =(x-3)(x+5) Summary: ① Factorization of X 2+(P+Q) X+PQ formula The characteristics of this kind of quadratic trinomial are: the coefficient of quadratic term is1; Constant term is the product of two numbers; The coefficient of a linear term is the sum of two factors of a constant term. So we can directly decompose some quadratic trinomials with coefficients of1:x2+(p+q) x+pq = (x+p) (x+q) 2kx2+MX+n into

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