2. Factorize x? -Really? + 10x + 25

3. Simplified post-assessment (1/2x+ 1/3y)? - ( 1/3x+ 1/2y)？ -(5/6x+5/6y) (1/6x-1/6y) where 2 = x? = 4 to the y power

4. (x-1) (n-65438+0 power of x+n-2 power of x+n-3 power of x+... +X+1) = n power of x-1Example: (x-/kloc-0) +x+ 1)=x to the fourth power.

According to this rule, 1+2+2? + 2? +2 to the 4th power +2 to the 5th power ... +2 to the 63rd power.

Step 5: Extract public factors.

12x square-12x square y-3x square y square

6. Variance formula

3ax quartic curve -3ay quartic curve

7. Complete square formula

25m2+64-80m2

8. Grouping decomposition

3xy-2x- 12y+8

9. Cross multiplication

X quartic -7x squared y squared +6y quartic

Score:

Add and subtract 5x/(x+y)+y/(x+y)

Multiply and divide b/(square of a-9) * (a+3)/(square of b -b)

Mixed braces a/(a-b)+b/(b-a) braces *ab/(a-b)

1. Factorization x3+2x2+2x+ 1

2. Factorizing A2B2-A2-B2+ 1

3. Try division to determine whether 15x2+x-6 is a multiple of 3x+2.

4. Is (1) 3x+2 a factor of 6x2+x-2? (Write the formula)

(2) If yes, please factorize 6x2+x-2.

5.A = 199 12, B = 99 12, (1) Find the value of A2-2AB+B2? (2) What is the value of A2-B2?

Is 6.2x+ 1 a factor of 4x2+8x+3? If yes, please factorize 4x2+8x+3.

7. Factorize (1) 3ax2-2x+3ax-2 (2) (x2-3x)+(x-3) 2+2x-6.

8. Let 6x2- 13x+k be a multiple of 3x-2, and find the value of k. ..

Is 9.3x a factor of x2? (Give reasons)

10. If -2x2+AX- 12 is divisible by 2x-3, find (1) a =? (2) Factorization -2x2+AX- 12.

1 1.( 1) Factorization AB-CD+AD-BC

(2) Use (1) to find1990× 29-/kloc-0 /× 765438+1990× 71-29×/kloc-.

12. Use the square difference formula to find 1992-992 =?

13. Find (67 12) 2-(32 12) 2 =?

14. Break down the following categories:

( 1)(2x+3)(x-2)+(x+ 1)(2x+3)(2)9 x2-66x+ 12 1

15. Please factorize 16x2-24x+9 with different factorization methods you have learned.

(1) Method 1: (2) Method 2:

16. Break down the following categories:

(1) 4x2-25 (2) x2-4xy+4y2 (3) Find A2-B2+2bc-C2 by the method of (1)(2).

17. Factorization

( 1)8 x2- 18(2)x2-(a-b)x-ab

18. Break down the following categories

( 1)9 x4+35 x2-4(2)x2-y2-2yz-z2

(3)a(b2-c2)-c(a2-b2)

19. Factorization (2x+1) (x+1)+(2x+1) (x-3)

20. Factorization 39x2-38x+8

2 1. Find the value of (65 12) 2-(34 12) 2 by factorization.

22. Decomposition A (B2-C2)-C (A2-B2)

23.a, B and C are integers. If A2+B2+C2+4A-8B- 14C+69 = 0, find the value of A+2B-3C.

24. Factorization 7 (x-1) 2+4 (x-1) (y+2)-20 (y+2) 2.

25. decompose xy2-2xy-3x-y2-2y- 1

26. Factorization 4x2-6ax+ 18a2

27. decompose 20a3bc-9a2bc-20ab3c

28. Factorization 2 x2-5x+2ax-5

29. Factorization 4x3+4x2-25x-25

30. Factorization (1-xy) 2-(y-x) 2

3 1. factorization

( 1)mx2-m2-x+ 1(2)a2-2ab+B2- 1

32. Break down the following categories

( 1)5x 2-45(2)8 1x 3-9x(3)x2-y2-5x-5y(4)x2-y2+2yz-z2

33. Factorization: xy2-2xy-3x-y2-2y- 1

34. Factorize Y2 (x-y)+Z2 (y-x)

35. Let x+ 1 be a factor of 2x2+AX-3, and find a =? (2) Find two roots of 2x2+AX-3 = 0

36.( 1) factorization x2+x+y2-y-2xy =?

(2) If (1) x-y = 99, what is the value of x2+x+y2-y-2xy?

75÷〔 138÷( 100-54)〕 85×(95- 1440÷24)

80400-(4300+870÷ 15) 240×78÷( 154- 1 15)

1437×27+27×563 〔75-( 12+ 18)〕÷ 15

2 160÷〔(83-79)× 18〕 280+840÷24×5

325÷ 13×(266-250) 85×(95- 1440÷24)

58870÷( 105+20×2) 1437×27+27×563

8 1432÷( 13×52+78) [37.85-(7.85+6.4)] ×30

156×[( 17.7-7.2)÷3] (947-599)+76×64

36×(9 13-276÷23) [ 192-(54+38)]×67

[(7. 1-5.6)×0.9- 1. 15]÷2.5 8 1432÷( 13×52+78)

5.4÷[2.6×(3.7-2.9)+0.62] (947-599)+76×64 60-(9.5+28.9)]÷0. 18 2.88 1÷0.43-0.24×3.5 20×[(2.44- 1.8)÷0.4+0. 15] 28-(3.4 / kloc-0/.25×2.4) 0.8×〔 15.5-(3.2 1 5.79)〕 (3 1.8 3.2×4)÷5 194-64.8÷ 1.8×0.9 36.72÷4.25×9.9 3.4 16÷(0.0 16×35) 0.8×[( 10-6.76)÷ 1.2]

( 136+64)×(65-345÷23) (6.8-6.8×0.55)÷8.5

0. 12× 4.8÷0. 12×4.8 (58+37)÷(64-9×5)

8 12-700÷(9+3 1× 1 1) (3.2× 1.5+2.5)÷ 1.6

85+ 14×( 14+208÷26) 120-36×4÷ 18+35

(284+ 16)×(5 12-8208÷ 18) 9.72× 1.6- 18.305÷7

4/7÷[ 1/3×(3/5-3/ 10)] (4/5+ 1/4)÷7/3+7/ 10

12.78-0÷( 13.4+ 156.6 ) 37.8 12-700÷(9+3 1× 1 1) ( 136+64)×(65-345÷23) 3.2×( 1.5+2.5)÷ 1.6

85+ 14×( 14+208÷26) (58+37)÷(64-9×5)

(6.8-6.8×0.55)÷8.5 (284+ 16)×(5 12-8208÷ 18)

0. 12× 4.8÷0. 12×4.8 (3.2× 1.5+2.5)÷ 1.6

120-36×4÷ 18+35 10. 15- 10.75×0.4-5.7

5.8×(3.87-0. 13)+4.2×3.74 347+45×2-4 160÷52

32.52-(6+9.728÷3.2)×2.5 87(58+37)÷(64-9×5)

[(7. 1-5.6)×0.9- 1. 15] ÷2.5 (3.2× 1.5+2.5)÷ 1.6

5.4÷[2.6×(3.7-2.9)+0.62] 12×6÷( 12-7.2)-6

3.2×6+( 1.5+2.5)÷ 1.6

5.8×(3.87-0. 13)+4.2×3.74

33.02-( 148.4-90.85)÷2.5

1, Public Welfare Law

If every term of a polynomial contains a common factor, then this common factor can be put forward, so that the polynomial can be transformed into the product of two factors.

Example 1, factorization factor x-2x-x (Huai' an senior high school entrance examination in 2003)

x -2x -x=x(x -2x- 1)

2. Application of formula method

Because factorization and algebraic expression multiplication have reciprocal relationship, if the multiplication formula is reversed, it can be used to decompose some polynomials.

Example 2, factorization factor a +4ab+4b (2003 Nantong senior high school entrance examination in 2003)

Solution: a +4ab+4b =(a+2b)

3. Grouping decomposition method

Factorizing the polynomial am+an+bm+bn, we can first divide the first two terms into a group and propose the common factor A, then divide the last two terms into a group and propose the common factor B, so as to get a(m+n)+b(m+n), and we can also propose the common factor M+N, so as to get (a+B) (m+).

Example 3. Decomposition factor m +5n-mn-5m

Solution: m +5n-mn-5m= m -5m -mn+5n.

= (m -5m )+(-mn+5n)

=m(m-5)-n(m-5)

=(m-5)(m-n)

4. Cross multiplication

For a polynomial in the form of mx +px+q, if a×b=m, c×d=q and ac+bd=p, the polynomial can be factorized into (ax+d)(bx+c).

Example 4, factorization factor 7x-19x-6

Analysis: 1 -3

7 2

2-2 1=- 19

Solution: 7x-19x-6=(7x+2)(x-3)

5. Matching method

For those polynomials that cannot be formulated, some can use it to make a completely flat way, and then factorize it with the square difference formula.

Example 5, Factorization Factor x +3x-40

Solution x +3x-40=x +3x+() -() -40

=(x+ ) -()

=(x++)(x++)

=(x+8)(x-5)

6. Removal and addition methods

Polynomials can be divided into several parts and then factorized.

Example 6: decomposition factor bc(b+c)+ca(c-a)-ab(a+b)

Solution: BC (B+C)+CA (C-A)-AB (A+B) = BC (C-A+A+B)+CA (C-A)-AB (A+B).

= BC(c-a)+ca(c-a)+BC(a+b)-ab(a+b)

=c(c-a)(b+a)+b(a+b)(c-a)

=(c+b)(c-a)(a+b)

7. Alternative methods

Sometimes in factorization, you can choose the same part of the polynomial, replace it with another unknown, then factorize it and finally convert it back.

Example 7, factorization factor 2x -x -6x -x+2

Solution: 2x-x-6x-x+2 = 2 (x+1)-x (x+1)-6x.

=x [2(x + )-(x+ )-6

Let y=x+, x [2(x+)-(x+ )-6.

= x [2(y -2)-y-6]

= x (2y -y- 10)

=x (y+2)(2y-5)

=x (x+ +2)(2x+ -5)

= (x +2x+ 1) (2x -5x+2)

=(x+ 1) (2x- 1)(x-2)

8. Root method

Let the polynomial f(x)=0 and find its roots as x, x, x, …x, ... x, then the polynomial can be decomposed into f (x) = (x-x) (x-x)...(x-x).

Example 8, factorization factor 2x +7x -2x-13x+6.

Solution: Let f(x)=2x +7x -2x-13x+6=0.

According to the comprehensive division, the roots of f(x)=0 are -3, -2, 1.

Then 2x+7x-2x-13x+6 = (2x-1) (x+3) (x+2) (x-1).

9. Mirror image method

Let y=f(x), make the image of function y=f(x), and find the intersection points x, x, x, …x, ... x between the image of function and the X axis, then the polynomial can be decomposed into f (x) = f (x) = (x-x) (x-x) ... (ten

Example 9, Factorization x +2x -5x-6

Solution: Let y= x +2x -5x-6.

Make it image, as shown on the right, and the intersection with the X axis is -3,-1, 2.

Then x+2x-5x-6 = (x+1) (x+3) (x-2).

10, principal component method

First, choose a letter as the main element, then arrange the items from high to low according to the number of letters, and then factorize them.

Example 10, factorization factor a (b-c)+b (c-a)+c (a-b)

Analysis: this question can choose a as the main element, arranged from high to low.

Solution: a (b-c)+b (c-a)+c (a-b) = a (b-c)-a (b-c)+(b c-c b)

=(b-c) [a -a(b+c)+bc]

=(b-c)(a-b)(a-c)

1 1, using the special value method.

Substitute 2 or 10 into x, find the number p, decompose the number p into prime factors, properly combine the prime factors, write the combined factors as the sum and difference of 2 or 10, and simplify 2 or 10 into x, thus obtaining factorization.

Example 1 1, factorization factor x +9x +23x+ 15.

Solution: let x=2, then x+9x+23x+15 = 8+36+46+15 =105.

105 is decomposed into the product of three prime factors, namely 105=3×5×7.

Note that the coefficient of the highest term in the polynomial is 1, while 3, 5 and 7 are x+ 1, x+3 and x+5, respectively, when x=2.

Then x+9x+23x+15 = (x+1) (x+3) (x+5).

12, undetermined coefficient method

Firstly, the form of factorization factor is judged, then the letter coefficient of the corresponding algebraic expression is set, and the letter coefficient is calculated, thus decomposing polynomial factor.

Example 12, decomposition factor x -x -5x -6x-4.

Analysis: It is easy to know that this polynomial has no primary factor, so it can only be decomposed into two secondary factors.

Solution: let x -x -5x -6x-4=(x +ax+b)(x +cx+d)

= x+(a+c)x+(AC+b+d)x+(ad+BC)x+BD

So the solution is

Then x-x-5x-6x-4 = (x+x+1) (x-2x-4

1- 14 x2

4x–2 x2–2

(x-y)3-(y-x)

x2–y2–x+y

x2–y2- 1(x+y)(x–y)

x2 + 1 x2 -2-( x - 1x )2

a3-a2-2a

4m2-9n2-4m+ 1

3a2+bc-3ac-ab

9 x2+2xy-y2

2x2-3x- 1

-2x2+5xy+2y2

10a(x-y)2-5b(y-x)

An+1-4an+4an- 1

x3(2x-y)-2x+y

x(6x- 1)- 1

2ax- 10ay+5by+6x

1-a2-ab- 14 b2

a4+4

(x2+x)(x2+x-3)+2

x5y-9xy5

-4x2+3xy+2y2

4a-a5

2x2-4x+ 1

4y2+4y-5

3X2-7X+2

8xy(x-y)-2(y-x)3

x6-y6

x3+2xy-x-xy2

(x+y)(x+y- 1)- 12

4ab-( 1-a2)( 1-b2)

-3 square meters -2 meters +4

a2-a-6

2(y-z)+8 1(z-y)

9m2-6m+2n-n2

ab(c2+d2)+cd(a2+b2)

a4-3a2-4

x4+4y4

a2+2ab+b2-2a-2b+ 1

x2-2x-4

4x2+8x- 1

2x2+4xy+y2

-m2–N2+2mn+ 1

(a+b)3d–4(a+b)2cd+4(a+b)c2d

(x+a)2-(x–a)2

–x5y–xy+2x3y

X6–x4–x2+ 1

(x+3)(x+2)+x2–9

(x–y)3+9(x–y)–6(x–y)2

(a2+B2– 1)2–4a2b 2

(ax+by)2+(bx–ay)2

x2+2ax–3 a2

3a3b2c-6a2b2c2+9ab2c3

xy+6-2x-3y

x2(x-y)+y2(y-x)

2x2-(a-2b)x-ab

a4-9a2b2

ab(x2-y2)+xy(a2-b2)

(x+y)(a-b-c)+(x-y)(b+c-a)

a2-a-b2-b

(3a-b)2-4(3a-b)(a+3b)+4(a+3b)2

(a+3)2-6(a+3)

(x+ 1)2(x+2)-(x+ 1)(x+2)2

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= .

(7)x2-9x+ 18= .

(8)2x2-5x-3= .

(9) 12x2-50x+8= .

40. Factorization (x+2) (x-3)+(x+2) (x+4) =.

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)-(x-6)= .

3(x+2)(x-5)-(x+2)(x-3)= .

(8)9x2+42x+49= .

( 1)(x+2)-2(x+2)2= .

(2)36x2+39x+9= .

(3)2x 2+ax-6x-3a = 1 .

(4)22x2-3 1x-2 1= .

70. Factorization 3ax2-6ax= =.

7 1. Factorization (x+ 1) x-5x =.

72. Factorization (2x+1) (x-3)-(2x+1) (x-5) =1

73. Factorization xy+2x-5y- 10 =

74. Factorization X2Y2-X2-Y2-6xy+4 =

x3+2x2+2x+ 1

a2b2-a2-b2+ 1

( 1)3ax2-2x+3ax-2

(x2-3x)+(x-3)2+2x-6

1)(2x+3)(x-2)+(x+ 1)(2x+3)

9x2-66x+ 12 1

17. Factorization

( 1)8 x2- 18(2)x2-(a-b)x-ab

18. Break down the following categories

( 1)9 x4+35 x2-4(2)x2-y2-2yz-z2

(3)a(b2-c2)-c(a2-b2)

19. Factorization (2x+1) (x+1)+(2x+1) (x-3)

20. Factorization 39x2-38x+8

2 1. Find the value of (65 12) 2-(34 12) 2 by factorization.

22. Decomposition A (B2-C2)-C (A2-B2)

24. Factorization 7 (x-1) 2+4 (x-1) (y+2)-20 (y+2) 2.

25. decompose xy2-2xy-3x-y2-2y- 1

26. Factorization 4x2-6ax+ 18a2

27. decompose 20a3bc-9a2bc-20ab3c

28. Factorization 2 x2-5x+2ax-5

29. Factorization 4x3+4x2-25x-25

30. Factorization (1-xy) 2-(y-x) 2

3 1. factorization

( 1)mx2-m2-x+ 1(2)a2-2ab+B2- 1

32. Break down the following categories

( 1)5x 2-45(2)8 1x 3-9x(3)x2-y2-5x-5y(4)x2-y2+2yz-z2

33. Factorization: xy2-2xy-3x-y2-2y- 1

34. Factorize Y2 (x-y)+Z2 (y-x)

1) factorization x2+x+y2-y-2xy =