(1) If the polynomial term has a common factor, then the common factor should be raised first; (2) If there is no common factor, try to decompose it by formula and cross multiplication; (3) If the above methods cannot be decomposed, you can try to decompose by grouping, splitting and adding items; (4) Factorization must be carried out until every polynomial factorization can no longer be decomposed. It can also be summarized in one sentence: "First, look at whether there is a common factor, and then look at whether there is a formula. Try cross multiplication, and group decomposition should be appropriate. " A few examples 1. Decomposition factor (1+y) 2-2x2 (1+y 2)+x 4 (1-y) 2. Solution: The original formula = (1+) x2 (1-y)-2x2 (1+y) (complement) = [(1+y)+x2 (1-y)]. Kloc-0/+y)+x 2 (1-y)-2x] = [(x+1) 2-y (x2-1)] [(x-1) 2-2 solution. When y is not equal to 0, x+3y, x+y, x-y, x+2y and x-2y are different from each other, and 33 cannot be divided into products of more than four different factors, so the original proposition holds. 3. The three sides A, B and C of 3.△ ABC have the following relationship: -C 2+A 2+2AB-2BC = 0. Prove that this triangle is an isosceles triangle. Analysis: This question is essentially factorizing the polynomial on the left side of the relation equal sign. Prove: ∫-C2+a2+2ab-2bc = 0, ∴ (a+c) (a-c)+2b (a-c) = 0. ∴ (a-c) (a+2b+c) = 0。 4. Factorization-12x2n× y n+18x (n+2) y (n+1)-6xn× y (n-1). Solution:-12x2n× y n+18x (n+2) y (n+1)-6xn× y (n-1) =-6xn× y (. The following examples can be used for reference: Example 1 Decomposition -A2-B2+2ab+4. Solution:-A2-B2+2AB+4 =-(A2-2AB+B2-4) =-(A-B+2) (A-B-2) The "negative" here means "negative sign". If the first term of a polynomial is negative, it is generally necessary to put forward a negative sign to make the coefficient of the first term in brackets positive. Prevent students from having two examples--9x2+4y2 = (-3x) 2-(2y) 2 = (-3x+2y) = (3x+2y)-12x2nyn+ 18xn. Solution:-12x2nyn+18xn+2yn+1-6xnyn-1=-6xnyn-1(2xny-3xy22+1) where If each term of a polynomial contains a common factor, first extract this common factor, and then further decompose this factor; "1" here means that when the whole term of the polynomial is a common factor, put forward this common factor first, and don't miss the 1 in brackets. Factorization must be carried out until each polynomial factor can no longer be decomposed. That is, break it down to the end, not give up halfway. The common factor contained in it should be "clean" at one time, leaving no "tail", and the polynomial in each bracket can not be decomposed again. Prevent students from making mistakes such as 4x4Y2-5x2Y2-9Y2 = Y2 (4x4-5x2-9) = Y2 (x2+1) (4x2-9). Attention should be paid to the exam: when there is no explanation for real numbers, it is generally enough to explain only rational numbers. If you have an explanation of real numbers, you usually turn to integers! From this point of view, the four attentions in factorization run through the four basic methods of factorization, which are in the same strain as the four steps of factorization or the four sentences of general thinking order: "First, see if there is a common factor, then see if a formula can be established, try cross multiplication, and group decomposition should be appropriate." Edit this paragraph and apply 1 to polynomial division. 2. It is applied to finding the roots of higher-order equations. 3. By the way, Mei Sen composite decomposition has made some insignificant progress: 1, p=4r+3. If 8r+7 is also a prime number, then: (8r+7) | (2 p- 1). That is, (2p+1) | (2p-1); For example: 23 | (211-1); ; 1 1=4×2+3; 47|(2^23- 1); ; 23=4×5+3; 167|(2^83- 1); ,,,.83=4×20+3; . . . . 2, p = 2n× 3 2+1,then (6p+ 1) | (2 p- 1), for example: 223 | (2 37-1); ; 37=2×2×3×3+ 1; 439|(2^73- 1); 73=2×2×2×3×3+ 1; 3463|(2^577- 1); ; 577=2×2×2×2×2×2×3×3+ 1; ,,,。 3, p = 2 n× 3 m× 5 s- 1, then (8p+1) | (2p-1); . For example; 233|(2^29- 1); 29=2×3×5- 1; ; 1433|(2^ 179- 1); 179=2×2×3×3×5- 1; 19 13|(2^239- 1); 239=2×2×2×2×3×5- 1; ,,,。 There are still some progress in the decomposition of Mason number, so I won't describe them one by one.