( 1)-2x5n- 1yn+4x3n- 1yn+2-2xn- 1yn+4； (2)x3-8y 3-z3-6x yz； (3)a2+B2+C2-2bc+2ca-2ab； (4) A7-A5B2+A2B5-B7。 Solution (1) The original formula =-2xn-1yn (x4n-2x2ny2+y4) =-2xn-1yn [(x2n) 2-2x2ny2+(. 2 (xn+y) 2。 (2) The original formula = x3+(-2y) 3+(-z) 3-3x (-2y) (-z) = (x-2y-z) (x2+4y2+z2+2xy+xz-2yz). The original formula = (A7-A5B2)+(A2B5-B7) = A5 (A2-B2)+B5 (A2-B2) = (A2-B2) (A5+B5) = (A+B) (A+B) (A4-A3B+A2B2) (2) (. (3)(x+ 1)4+(x2- 1)2+(x- 1)4； (4) A3b-AB3+A2+B2+ 1。 The solution (1) decomposes -3 into-1- 1. Original formula = X9+X6+X3- 1. =(x3- 1)(X6+x3+ 1)+(x3+ 1)(x3+ 1)+(x3- 1)=(x3- 1)(X6+2 x3+3)=(x- 1)(x2+x+ 1)(X6+2 x3+3)。 (2) decompose 4mn into 2mn+2mn. Original formula = (m2-1) (N2-1)+2mn+2mn = m2n2-m2-N2+1+2mn = (m2n2+2mn+1). (Mn-M+N+ 1)。 (3) Break (x2- 1)2 into 2 (x2- 1) 2. Original formula = (x+ 1). 4+2 (x+1) 2 (x-1) 2+(x-1) 4]-(x2-1) 2 = (x+1) 2+(x- The original formula = A3b-AB3+A2+B2+1+AB-AB = (A3b-AB3)+(A2-AB)+(AB+B2+1) = AB (A+B) (A-B). +(a b+B2+ 1)=[a(a-b)+ 1](a b+B2+ 1)=(a2-a b+ 1)(B2+a b+ 1)。 ( 1)-2x5n- 1yn+4x3n- 1yn+2-2xn- 1yn+4； (2)x3-8y 3-z3-6x yz； (3)a2+B2+C2-2bc+2ca-2ab； (4) A7-A5B2+A2B5-B7。 Solution (1) The original formula =-2xn-1yn (x4n-2x2ny2+y4) =-2xn-1yn [(x2n) 2-2x2ny2+(. 2 (xn+y) 2。 (2) The original formula = x3+(-2y) 3+(-z) 3-3x (-2y) (-z) = (x-2y-z) (x2+4y2+z2+2xy+xz-2yz). Equation (5) can be directly used to solve the problem as follows: original equation = a2+(-b) 2+C2+2 (-b) c+2ca+2a (-b) = (a-b+c) 2 (4) original equation = (A7-A5B2)+(A2B5-B7) = A5. =(a+b)2(a-b)(a4-a3 b+a2 B2-ab3+B4)( 1)x9+X6+x3-3； (2)(m2- 1)(N2- 1)+4mn； (3)(x+ 1)4+(x2- 1)2+(x- 1)4； (4) A3b-AB3+A2+B2+ 1。 The solution (1) decomposes -3 into-1- 1. Original formula = X9+X6+X3- 1. =(x3- 1)(X6+x3+ 1)+(x3+ 1)(x3+ 1)+(x3- 1)=(x3- 1)(X6+2 x3+3)=(x- 1)(x2+x+ 1)(X6+2 x3+3)。 (2) decompose 4mn into 2mn+2mn. Original formula = (m2-1) (N2-1)+2mn+2mn = m2n2-m2-N2+1+2mn = (m2n2+2mn+1). (Mn-M+N+ 1)。 (3) Break (x2- 1)2 into 2 (x2- 1) 2. Original formula = (x+ 1). 4+2 (x+1) 2 (x-1) 2+(x-1) 4]-(x2-1) 2 = (x+1) 2+(x- The original formula = A3b-AB3+A2+B2+1+AB-AB = (A3b-AB3)+(A2-AB)+(AB+B2+1) = AB (A+B) (A-B). +(ab+B2+1) = [a (a-b)+1] (ab+B2+1) = (a2-ab+1) (4 .10000.000000000016 (2)x4- 1 1x2y 2+y2； (3)x3+9 x2+26x+24； (4) x4- 12x+323.3。 Decomposition factor: (1) (2x2-3x+1) 2-22x2+33x-1; (2)x4+7x 3+ 14x 2+7x+ 1； (3)(x+y)3+2xy( 1-x-y)- 1； (4) (X+3) (X2- 1) (X+5)-20。 First, multiple-choice questions 1. It is known that y2+my+ 16 is completely flat. The value of m is () a.8b.4c.8d.42 The following polynomials can be factorized by the complete square formula: () a.x2-6x-9b.a2-16a+32c.x2-2xy+4Y2D.4a2-4a+65438+. 0+2x)2b . 6a-9-a2 =-(a-3)2c . 1+4m-4 m2 =( 1-2m)2d . x2+xy+y2 =(x+y)2 4。 Put x4-2x2xy2+y4. The result is () a. (x-y) 4b. (x2-y2) 4c. [(x+y) (x-y)] 2d. (x+y) 2 (x-y) 22. Fill in the blanks. It is known that 9x2-6xy+k is completely flat. Then the value of k is _ _ _ _ _ _ _ .6.9a2+(_ _ _ _)+25b2 = (3a-5b) 27. -4x 2+4xy+(_ _ _ _ _ _)=-(_ _ _ _ _ _)。 Then the value of a is _ _ _ _ _ _ _. Third, solve the problem 9. The following factors are decomposed:1a2+10a+252m2-12mn+36n23xy3-2x2y2+x3y4 (x2+4y2) 2-65438+. Find the value of algebraic expression 4x2+ 12xy+9y2. 1 1. It is known that │x-y+ 1│ and x2+8x+ 16 are reciprocal. Find the value of x2+2xy+y2. When solving a problem, if you focus on the whole problem, think, associate and explore from all directions, think and deform as a whole, and determine the solution strategy from different aspects, the problem can be solved quickly. Can you factorize the following formula with holistic thinking method? ① (x+2y) 2-2 (x+2y)+1② (a+b) 2-4 (a+b-1) Reference answer:1.c2.d3.b4.d5.y26-30a. 2x-y 8。 -2 or-12 9. ①(a+5)2； ②(m-6n)2； ③xy(x-y)2； ④(x+2y)2(x-2y)2 10.4 1 1.49 12。 ①(x+2y- 1)2； ②(a+b-2)2 1.2(a-3)(a-3)-a+3 2.9(m+n)(m+n)- 16(m-n)(m-n)3. 15(a-2)-9b(a- 1)(2-a)4. 16a×a×a×a-72a×a×b×b+8 1 b×ban+ 1-4an+4an- 1(3). x3(2x-y)-2x+y(4)。 x(6x- 1)- 1(5). 2ax- 10ay+5by+6x(6). 1-a2-a b- 14 B2 *(7). a4+4(8)。 (x2+x)(x2+x-3)+2(9). x5y-9xy 5( 10)。 -4x 2+3xy+2 y2( 1 1). 4a-a5( 12). 2x 2-4x+ 1( 13). 4 y2+4y-5( 14)3 x2-7X+2 、- m2–N2+2mn+ 1 2 、( a+b)3d–4(a+b)2cd+4(a+b)(x+a)2 –( x–a)2 4。 5.–x5y–xy+2x3y 6。 X6–x4–x2+ 1 7。 (x+3)(x+2)+x2–9 8。 (x–y)3+9(x–y)–6(x–y)2 a2bm+3-2 abm+2+BM+ 1( 17)M4+4 m2-5( 18)-A2+ 1+2ab-B2( 19)(x2+7x+2)2- 16(20)(a b+ / Kloc-0/)2-(. 1, factorization: 9x2-1= _ _ _ _ _ _ _ _ _ _ _ _, 4x2-4x+1= _ _ _ _ _ _ _ _ _ _. If an+2-an = _ _ _ _ _ _ _ _ _ 2 and the polynomial x2+MX+36 are completely flat, then m = _ _ _ _ _ _ _ and the polynomial x2+ax+b can be factorized. If the value of the polynomial x3-4x2-9x+m is 0, then m = _ _ _ _ _ _ _ _ _ _ _ _ _ The result of polynomial factorization is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _. _ factorization is ............................ () (a) (a+3) = a2-9 (b) 4a2+4a+3 = (2a+1) 2+2 (c) x2-. 1, the number of polynomials that can be completely square decomposed is the correct decomposition number ............................ ()13xy+6y2-x-2y = (3xy-x)+(6y2-2y) 23xy+6y2-x-2y = (3xy+6y2)- (ax+by)2+(bx–ay)2) 1。 Which of the following polynomials contains a factor of 2x+3 (1) 2x+3+3 (2) 4x2-9? ( 1)(x-6)(2)(x+7)(3)(2x-3)(4)(2x+3)(3)。 A × C+B× C ( 1) A+B × C(？ () 4. Which of the following is not a factor of x2-4? (1) x+2x-2 (3) x2-4 (4) x2. () 5. Which of the following cannot be a factor of A2-B2? (1) A2+B2 (2) A+B (3) A-B (4) A2-B2. () 6. Which of the following is wrong? (1) (-a+b) 2 = a2-2ab+B2 (2) (a-b) (a+b) = a2-B2 (3) (a-b) 2 = a2-2ab-B2 (4) (4+3) 2 = (1) 2x-3 (2) x+7 (3) x-7 (4) 2x+7. () 8. Which of the following is the common factor of 2x2+3x+ 1 and 4x2-4x-3? (1) x+1(2) x+2 (3) 2x-3 (4) 2x+1.() 9. Factorization (a+2) 2-3 (a+2) = (/kloc-) Which of the following is true? (1) A2-B2 = (a-b) 2 (2) A2-2ab+B2 = (a+b) (3) A2+2ab+B2 = (a+b) 2 (4) A2+B2 = (a+b) () 12. if 5x2-7x-6 = (5x+A) (x+B), then (1) a =-3 (2) b =-2 (3) ab = 6 (4) a+b = 5. () 1)a>03.x2+MX+n = (x+a) (x+b), if m 0, then (1) a > 0, b > 0 (2) a < 0, b < 15x-2 (2)15x+2 (3) 3x-1(4) 3x+1.()15. Which of the following is (x-4) (x-5? (1) x-2x+11(3) x-1(4) x+3. ()16. If 6x2-25x+4 = (1) ABCD = 25 (2) A+B+C+D = 24 (3) If A = 1, CD = 6 (4) If A = 1, D =- 1. () 17.4a2- 1 is equal to the following formula? ( 1)(4a- 1)2(2)(2a- 1)2(3)(4a+ 1)(4a- 1)(4)(2a+ 1)(2a- 1)。 () 18.x2+y2 equals (1) (x+y) 2 (2) (x+y) 2+2xy (3) (x-y) 2+2xy (4) (x-y) 2-2xy. () 19. You can use two squares with side length of xcm, nine rectangles with length and width of x, 1cm and four squares with side length of 1cm to spell out a square with length of (x+4) cm and width of (1) (2x+/kl. () 20. Which of the following is the factor of 2x2+3x+ 1 and 4x2-4x-3? (1) 2x-1(2) 2x+1(3) 2x-3 (4) x+1.() 21.Which of the following formulas is not a factor of 9X2-25? (1) 3x+5 (2) 3x-5 (3) 9x+5 (4) 9x2-25. () 22. Factorization x2-3x+2 = (x+a) (a+b) Then (1) a+. () 23. Which of the following quadratic factors is X- 1? (1) x2+5x+6 (2) x2-5x-6 (3) x2+5x-6 (4) x2-5x+6. () 24. (-x+y) 2 equals (1)-(x-y). () 25. If x+y =-5 and x-y = 15, then x2-y2 = (1)-5 (2)-1(3)-15 (4)/kloc. () 26.x2+px+q = (x+a) (x+b), if a < 0 and b < 0, then (1) p > 0 (2) q < 0 (3) pq > 0 (4) q > 0. () 27. If (x-5) 2-(x-5)- 12 can be decomposed into (x+a) (x+b), then a+b is equal to (1)-1(2). () 28.ax-CX-by+cy+bx-ay can be divided into the following types? (1) (x-y) (a-b-c) (2) (x+y) (a+b-c) (3) (x-y) (a-b+c) (4) (x-y) (a+b-c) .. (1) x2+2ax+x = x (x+2a) (2) 2x2-8 = x2-4 = (x-2) (x+2) (3) 36x2-84x+49 = (7-6x) 2 (4) x2-. If 2x3+3x2+MX+ 1 is a multiple of X+ 1, then m = 2. Factorization 3a3b2c-6a2bc2+9ab2c3 = 3. Factorization xy+6-2x-3y = 4. Factorization x2 (x-y)+y2 (y-x) = 5. Factorization 2x2-(a-2b) x-ab = 6. Try to decompose x3+3x2-4 = 8. Factorization AB (x2-y2)+XY (A2-B2) = 9. Factorization (x+y) (a-b-c)+(x-y) (b+c-a) =10. Factorization A2. 2 = 12. Factorization (a+3) 2-6 (a+3) = 13. Factorization (x+1) 2 (x+2)-(x+1) (x+2). 15. Using the square difference formula, find the standard decomposition formula 489 1 =. 16.2x+ 1 is a factor of 4x2+5x- 1 A: Yes. 17. If 6x2-7x+m is a multiple of 2x-3, the common factor of m = 18. X2+2x+ 1 and x2- 1 Yes. 19. If x+2 is a factor of x2+kx-8, find k =. 20. If 4x2+8x+3 is a multiple of 2x+ 1, please factorize 4x2+8x+3 =. 2 1.2x+ 1 is a factor of 4x2+8x+3. Please factorize 4x2+8x+3 =. 22. The factorial of (1) x+2 (2) x+4 (3) x+6 (4) x-6 (5) x2+2x3+24 is listed in X2-2x-24 (all pairs are given points) 23. Factorial decomposition of the following categories: (65438) (2)16x2-81=. (3) 9x2-30x+25 =. (4) x2-7x-30 = .24. If x2+ax-/kloc-0. 25. Please fill in the appropriate number: x2- 16x+= (x-) 2. 26. decompose the following categories: (1) xy-xz+x =; (2)6(x+ 1)-y(x+ 1)=(3)x2-5x-px+5p =； (4) 15 x2- 1 1x- 14 = 27。 Let 7x2- 19x-6 = (7x+a) (bx-3), where a and b are integers, then 2A+B = 29. Calculate the value of (1.99) 2-4× 1.99+4 as follows. 30. If x2+AX- 12 can be decomposed into (x+6) (x+b) and both a and b are integers, then a+b =. 3 1. It is known that 9X2-MX+25 = (3x-n) 2 and n is a positive integer, then m+n =. 32. If 2x3+11x2+18x+9 = (x+1) (ax+3) (x+b), then A-b =. 33.2992-3992 = 34. Fill in the appropriate numbers to make it completely flat 4x2-20x+. 35. Factorization x2-25 =. 36. Factorization x2-20x+ 100 =. 37. Factorization x2+4x+3 =. 38. Factorization 4x2- 12x+5 =. 39. Break down the following categories: (1) 3ax2-6ax =. (2) x (x+2)-x =. (3) x2-4x-ax+4a =. (4) 25x2-49 =. (5) 36x2-60x+25 =. (6) 4x2+12x+9 =. (. 4 1. Factorization 2ax2-3x+2ax-3 =. 42. Factorize 9X2-66x+ 12 1 =. 43. Factorization 8-2x2 =. 44. Factorization x2-x+ 14 =. 45. Factorization 9X2-30x+25 =. 46. Factorization -20x2+9x+20 =. 47. Factorization 12x2-29x+ 15 =. 48. Factorization is 36x2+39x+9 =. 49. Factorization 2 1x2-3 1x-22 =. 50. Factorization 9x4-35x2-4 =. 5 1. Factorization (2x+1) (x+1)+(2x+1) (x-3) =. 52. Factorization 2ax2-3x+2ax-3 =. 53. Factorize X (y+2)-X-Y- 1 =. 54. Factorization (x2-3x)+(x-3) 2 =. 55. Factorize 9X2-66x+ 12 1 =. 56. Factorization 8-2x2 =. 57. Factorize x4- 1 =. 58. Factorization x2+4x-xy-2y+4 =. 59. Factorization 4x2- 12x+5 =. 60. Factorization 2 1x2-3 1x-22 =. 6 1. Factorization 4x2+4xy+y2-4x-2y-3 =. 62. Factorization 9X5-35x3-4x =. 63. Break down the following categories: (1) 3x2-6x =. (2) 49x2-25 =. (3) 6x2-13x+5 =. (4) x2+2-3x =. (5)12x2-23x-24 =. (6) (x+6) (x-6) 65. If x2+MX- 15 can be decomposed into (x+n) (x-3) and both m and n are integers, then m = n =. 66. Find the sum, difference, product or quotient of the following. ( 1)(65 12 )2-(34 12 )2= 。 (2)(79 13 )2+2×79 13 ×23 +49 = 。 (3) 1998×0.48-798×0.48-798×0.52+ 1998×0.52= 。